Dividing Polynomials and Powers of Binomials

Think About It!

• To check the answer, one could use long division to verify the quotient.

Step 5

• Write the quotient: 3x^2 - 11x - 20. The degree of the dividend is 3, and the degree of the divisor is 1, so the degree of the quotient is 2.
• The final sum in the synthetic division is 0, so the remainder is 0.

Watch Out!

• When terms are missing from the polynomial, add placeholders for those terms.
• In the example shown, there are 0x^3 terms.

Example 5. Divisor with a Coefficient Other Than 1

• Problem: Find (4x^3 - 7x^2 + 4x + 9) \div (2x - 1)
• To use synthetic division, the lead coefficient of the divisor must be 1.
• Divide the numerator and the denominator by 2:
  \frac{4x^3 - 7x^2 + 4x + 9}{2x - 1} = \frac{(4x^3 - 7x^2 + 4x + 9) \div 2}{(2x - 1) \div 2} = \frac{2x^3 - \frac{7}{2}x^2 + 2x + \frac{9}{2}}{x - \frac{1}{2}}
• x - a = x - \frac{1}{2}, therefore a = \frac{1}{2}.
• Complete the synthetic division:
  \begin{array}{c|cccc}
  \frac{1}{2} & 2 & -\frac{7}{2} & 2 & \frac{9}{2} \
  & & 1 & -\frac{5}{4} & \frac{3}{8} \
  \hline
  & 2 & -\frac{5}{2} & \frac{3}{4} & \frac{39}{8}
  \end{array}
• The resulting expression is 2x^2 - \frac{5}{2}x + \frac{3}{4} + \frac{\frac{39}{8}}{x - \frac{1}{2}}.
• Simplify the fraction:
  \frac{\frac{39}{8}}{x - \frac{1}{2}} = \frac{\frac{39}{8}}{\frac{2x - 1}{2}} = \frac{39}{8} \cdot \frac{2}{2x - 1} = \frac{39}{4(2x - 1)}
• The solution is 2x^2 - \frac{5}{2}x + \frac{3}{4} + \frac{39}{4(2x - 1)}.
• Check the answer by using long division.

Check

• Find (4x^4 + 3x^3 - 12x^2 - x + 6) \div (4x + 3)

Practice

Example 1

• Simplify each expression:
  1. \frac{15y^3 + 6y^2 + 3y}{3y}
  2. (45a^5 - 64a^4 + 12a^3 - 8a^2) \div (4a^2)^{-1}
  3. (6j^2k - 9jk^2) \div (3jk)
  4. (4a^2h^2 - 8a^3h + 3ah^3) \div (2ah^2)

Examples 2 and 3

• Simplify by using long division:
  5. (n^2 + 7n + 10) \div (n + 5)
  6. (d^2 + 4d + 3) \div (d + 1)^{-1}
  7. (2t^2 + 13t + 15) \div (t + 6)
  8. (6y^2 + y - 2) \div (2y - 1)^{-1}
  9. (4g^2 - 9) \div (2g + 3)
  10. (2x^2 - 5x - 4) \div (x - 3)

Examples 4 and 5

• Simplify using synthetic division:
  11. (3v^2 - 7v - 10) \div (v - 4)^{-1}
  12. (3t^4 + 4t^3 - 32t^2 - 5t - 20) \div (t + 4)^{-1}
  13. \frac{y^3 + 6}{y + 2}
  14. \frac{2x^3 - x^2 - 18x + 32}{2x - 6}
  15. (4p^3 - p^2 + 2p) \div (3p - 1)
  16. (3c^4 + 6c^3 - 2c + 4) \div (c + 2)^{-1}

Mixed Exercises

• Simplify the following:

  17. (m^2 + m - 6) \div (m + 4)
  18. (a^3 - 6a^2 + 10a - 3) \div (a - 3)
  19. (2x^3 - 7x^2 + 7x - 2) \div (x - 2)
  20. (x^3 + 2x^2 - 34x + 9) \div (x + 7)
  21. (x + 8) \div (x + 2)
  22. (6x^3 + x^2 + x) \div (2x + 1)
  23. (28c^3d^2 - 21cd^2) \div (14cd)
  24. (x - y) \div (x - y)
  25. \frac{n^3 + 3n^2 - 5n - 4}{n + 4}
  26. (a^3b^2 - a^2b + 2b) \div (-ab)^{-1}
  27. \frac{3z^5 + 5z^4 + z + 5}{z + 2}
  28. \frac{p^3 + 2p^2 - 7p - 21}{p + 3}

• 29. STRUCTURE: Jada used long division to divide x^4 + x^3 + x^2 + x + 1 by x + 2. Her work is shown below with three terms missing. What are A, B, and C?
  \begin{array}{r} x^3 - x^2 + 3x - 5 \
  x + 2 \overline{\smash)x^4 + x^3 + x^2 + x + 1} \
  \underline{-(x^4 + 2x^3)} \downarrow \
  -x^3 + A \
  \underline{-(-x^3 - 2x^2)} \downarrow \
  3x^2 + x \
  \underline{-(3x^2 + B)} \downarrow \
  -5x + 1 \
  \underline{-(-5x - 10)} \
  C
  \end{array}

• 30. AVERAGES: Bena has a list of n + 1 numbers and she needs to find their average. Two of the numbers are n^3 and 2. Each of the other n - 1 numbers are all equal to 1. Find the average of these numbers.

• 31. VOLUME: The volume of a cylinder is x^3 + 32x^2 - 304x + 640. If the height of the cylinder is x + 40 feet, find the area of its base in terms of x and \pi.

• 32. REASONING: Rewrite \frac{6x^4 + 2x^3 - 16x^2 + 24x + 32}{2x + 4} as q(x) + \frac{r(x)}{d(x)} using long division. What does the remainder indicate in this problem?

• 33. CONSTRUCT ARGUMENTS: Determine whether you have enough information to fill in the missing pieces of the long division exercise shown. If so, copy and complete the long division. Justify your response.
  \begin{array}{r}
  3x + 0 \
  9x^2 + 0 \
  9x^2 + 3x \
  -3x + 5
  \end{array}

• 34. REGULARITY: Rewrite \frac{2x^5 - 7x^4 - 15x^3 + 2x^2 + 3x + 6}{2x + 3} as q(x) + \frac{r(x)}{g(x)} using long division.

  • a. Identify q(x), r(x), and g(x).
  • b. How can you check your work using the expressions of q(x), g(x), and r(x)?

• 35. STRUCTURE: When a polynomial is divided by 4x - 6, the quotient is 2x^2 + x + 1 and the remainder is -4. What is the dividend, f(x)? Explain.

• 36. USE A MODEL: Luciano has a square garden. A new garden will have the same width and a length that is 3 feet more than twice the width of the original garden.

  • a. Define a variable. Copy the diagrams. Label each side of the diagrams with an expression for its length.
  • b. Write a ratio to represent the percent increase in the area of the garden. Use polynomial division to simplify the expression.
  • c. Use your expression from part b to determine the percent of increase in area if the original garden was a 12-foot square. Check your answer.

• 37. REGULARITY: Mariella makes the following claims about the degrees of the polynomials in \frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}. Do you agree with each claim? Justify your answers and provide examples.

  • a. The degree of d(x) must be less than the degree of f(x).
  • b. The degree of r(x) must be at least 1 less than the degree of d(x).
  • c. The degree of q(x) must be the degree of f(x) minus the degree of d(x).

Higher-Order Thinking Skills

• 38. FIND THE ERROR: Tomo and Jamal are dividing 2x^3 - 4x^2 + 3x - 1 by x - 3. Tomo claims that the remainder is -100. Jamal claims that the remainder is 26. Is either of them correct? Explain your reasoning.

• 39. PERSEVERE: If a polynomial is divided by a binomial and the remainder is 0, what does this tell you about the relationship between the binomial and the polynomial?

• 40. ANALYZE: What is the relationship between the degrees of the dividend, the divisor, and the quotient in any polynomial division exercise?

• 41. CREATE: Write a quotient of two polynomials for which the remainder is 3.

• 42. WRITE: Compare and contrast dividing polynomials using long division and using synthetic division.

• 43. PERSEVERE: Mr. Collins has his class working with bases and polynomials. He wrote on the board that the number 1111 in base B has the value B^3 + B^2 + B + 1. The class was then given the following questions to answer.

  • a. The number 11 in base B has the value B + 1. What is 1111 (in base B) divided by 11 (in base B)?
  • b. The number 111 in base B has the value B^2 + B + 1. What is 1111 (in base B) divided by 111 (in base B)?

Lesson 2-5 Powers of Binomials

Explore Expanding Binomials

• Use online interactive tool to complete the Explore activity.
• INQUIRY: How can you use Pascal's triangle to write expansions of binomials?

Learn Powers of Binomials

• Expand binomials by following rules and using patterns.

Key Concept: Binomial Expansion

• In the binomial expansion of (a + b)^n:
  • There are n + 1 terms.
  • n is the exponent of a in the first term and of b in the last term.
  • In successive terms, the exponent of a decreases by 1, and the exponent of b increases by 1.
  • The sum of the exponents in each term is n.
  • The coefficients are symmetric.

Pascal's Triangle

• A triangle of numbers where a row represents the coefficients of an expanded binomial (a + b)^n.
• Each row begins and ends with 1.
• Each coefficient is found by adding the two coefficients above it in the previous row.

Binomial Theorem

• Used instead of writing out the rows of Pascal's triangle to expand a binomial.
• Uses combinations to calculate the coefficients of the binomial expansion.

Key Concept: Binomial Theorem

• If n is a natural number, then (a + b)^n = C0a^n b^0 + C1a^{n-1}b^1 + C2a^{n-2}b^2 + C3a^{n-3}b^3 + … + C_na^0b^n
• Can also be written as:
  \frac{n!}{0!(n-0)!}a^n b^0 + \frac{n!}{1!(n-1)!}a^{n-1}b^1 + \frac{n!}{2!(n-2)!}a^{n-2}b^2 + \frac{n!}{3!(n-3)!}a^{n-3}b^3 + … + \frac{n!}{n!(n-n)!}a^0b^n

Example 1 Use Pascal's Triangle

• Use Pascal's triangle to expand (x + y)^6.
• (x + y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6

Think About It!

• Both C0 and Cn equal 1. This means that the first and the last terms of a binomial expansion have a coefficient of 1.
• This relates to Pascal's triangle because each row begins and ends with 1.

Study Tip

• Combinations: C_r refers to the number of ways to choose r objects from n distinct objects. In the Binomial Theorem, n is the exponent of (a + b)^n, and r is the exponent of b in each term.

• To calculate the coefficients, remember that n! represents n factorial. This is the product of all counting numbers beginning with n and counting backward to 1. For example, 3! = 3 \cdot 2 \cdot 1 = 6.

• 0! is defined as 1.

• (x + y)^0 = 1

• (x + y)^1 = 1x + 1y

• (x + y)^2 = 1x^2 + 2xy + 1y^2

• (x + y)^3 = 1x^3 + 3x^2y + 3xy^2 + 1y^3

• (x + y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1y^4

• (x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

• (x + y)^6 = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6

• (x + y)^7 = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7

Talk About It!

• A shortcut to writing out rows of Pascal's triangle is recognizing that the triangle is symmetric, so you only need to calculate half of the numbers in each row before mirroring them. Also, each number is the sum of the two numbers directly above it.

Check

• Write the expansion of (c + d)^3.
• (c + d)^3 = c^3 + 3c^2d + 3cd^2 + d^3

Example 2 Use the Binomial Theorem

• BASEBALL: In 2016, the Chicago Cubs won the World Series for the first time in 108 years. During the regular season, the Cubs played the Atlanta Braves 6 times, winning 3 games and losing 3 games. If the Cubs were as likely to win as to lose, find the probability of this outcome by expanding (w + l)^6.
• (w + l)^6 = 6C0w^6 + 6C1w^5l + 6C2w^4l^2 + 6C3w^3l^3 + 6C4w^2l^4 + 6C5wl^5 + 6C6l^6
  = \frac{6!}{6!0!}w^6 + \frac{6!}{5!1!}w^5l + \frac{6!}{4!2!}w^4l^2 + \frac{6!}{3!3!}w^3l^3 + \frac{6!}{2!4!}w^2l^4 + \frac{6!}{1!5!}wl^5 + \frac{6!}{0!6!}l^6
  = w^6 + 6w^5l + 15w^4l^2 + 20w^3l^3 + 15w^2l^4 + 6wl^5 + l^6
• By adding the coefficients, you can determine that there were 64 combinations of wins and losses that could have occurred.
• 20w^3l^3 represents the number of combinations of 3 wins and 3 losses. Therefore, there was a \frac{20}{64} or about a 31% chance of the Cubs winning 3 games and losing 3 games against the Braves.

Study Tip

• Assumptions: To use the Binomial Theorem, we assumed that the teams had an equal chance of winning and losing. Assuming that allows us to reasonably estimate the probability of an outcome with only the coefficient.
• To find probabilities of events that are not equally likely, substitute the probability of each event for a and b in the expansion of (a + b)^n.

Check

• GAME SHOW: A group of 8 contestants are selected from the audience of a television game show. If there are an equal number of men and women in the audience, find the probability of the contestants being 5 women and 3 men by expanding (w + m)^8. Round to the nearest percent if necessary.
• The probability is approximately 22%.

Example 3 Coefficients Other Than 1

• Expand (2c - 6d)^4.
• (2c - 6d)^4 = 4C0(2c)^4 + 4C1(2c)^3(-6d) + 4C2(2c)^2(-6d)^2 + 4C3(2c)(-6d)^3 + 4C4(-6d)^4
  = 1(16c^4) + 4(8c^3)(-6d) + 6(4c^2)(36d^2) + 4(2c)(-216d^3) + 1(1296d^4)
  = 16c^4 - 192c^3d + 864c^2d^2 - 1728cd^3 + 1296d^4

Study Tip

• Coefficients: When the binomial to be expanded has coefficients other than 1, the coefficients will no longer be symmetric. In these cases, it may be easier to use the Binomial Theorem.

Practice

Example 1

• Use Pascal's triangle to expand each binomial:
  1. (x - y)^3
  2. (a + b)^4
  3. (g - h)^4
  4. (m + 1)^4
  5. (y - z)^6
  6. (d + 2)^8

Example 2

• 7. BAND: A school band went to 4 competitions during the year and received a superior rating 2 times. If the band is as likely to receive a superior rating as to not receive a superior rating, find the probability of this outcome by expanding (s + n)^4. Round to the nearest percent if necessary.
• 8. BASKETBALL: Oliver shot 8 free throws at practice, making 6 free throws and missing 2 free throws. If Oliver is equally likely to make a free throw as he is to miss a free throw, find the probability of this outcome by expanding (m + n)^8. Round to the nearest percent if necessary.

Example 3

• Expand each binomial:
  9. (3x + 4y)^5
  10. (2c - 2d)^7
  11. (8h - 3)^4
  12. (4a + 3b)^6

Mixed Exercises

• Expand each binomial:
  13. (x + 1)^5
  14. (x - 1)^6
  15. (2b + 1)^5
  16. (3c + \frac{1}{10})^3
• 17. STRUCTURE: Out of 12 frames, Vince bowled 6 strikes. If Vince is as likely bowl a strike as to not bowl a strike in one frame, find the probability of this outcome. Round to the nearest percent if necessary.
• 18. REGULARITY: A group of 10 choir members are selected at random to perform solos. If there are an equal number of boys and girls in the choir, find the probability of the choir members selected being 7 boys and 3 girls. Round to the nearest percent if necessary.
• 19. USE A MODEL: A company is developing a robotic welder that produces circuit boards. At this stage in its development, the robotic welder only produces 50% of the circuit boards correctly. Use the Binomial Theorem to find the probability that 5 of 7 circuit boards chosen at random are correct.
• 20. USE A MODEL: Diego flips a fair coin 12 times. What is the probability that the coin lands on tails 3 times? 5 times? 9 times?
• 21. REASONING: A test consists of 10 true-false questions. Matthew forgets to study and must guess on every question. What is the probability that he gets 8 or more correct answers on the test? Show your work using Pascal's Triangle.
• 22. REGULARITY: Use Pascal's Triangle to find the fourth term in the expansion of (2x + 7)^6. Why is it the same as the fourth term in the expansion of (7 + 2x)^6?
• 23. USE A SOURCE: Research the number of judges on the Supreme Court. For most rulings, a majority is needed. How many combinations of votes are possible for a majority to be reached?
• 24. STRUCTURE: Find the term in (a + b)^{12} where the exponent of a is 5.
• 25. PRECISION: Use the first four terms of the binomial expansion of (1 + 0.02)^{10} to approximate (1.02)^{10}. Evaluate (1.02)^{10} using a calculator and compare the value to your approximation.

Higher-Order Thinking Skills

• 26. PERSEVERE: Find the sixth term of the expansion of (\sqrt{a} + \sqrt{b})^{12}.
• 27. ANALYZE: Explain how the terms of (x + y)^n and (x - y)^n are the same and how they are different.
• 28. REGULARITY: Each row of Pascal's triangle is like a palindrome. That is, the numbers read the same left to right as they do right to left. Explain why this is the case.
• 29. CREATE: Write a power of a binomial for which the second term of the expansion is 6x^5y.
• 30. WRITE: Explain how to write out the terms of Pascal's triangle.

• Polynomials and polynomial functions can be used to model situations where quantities increase or decrease in a nonlinear pattern.

Lessons 2-1 and 2-2 Polynomial Functions and Graphs

• A power function is any function of the form f(x) = ax^n, where a and n are nonzero real numbers. The leading coefficient is a and the degree is n.
• Odd-degree functions will always have at least one real zero.
• Even-degree functions may have any number of real zeros or no real zeros at all.
• A polynomial function is a continuous function that can be described by a polynomial equation in one variable.
• The degree of a polynomial function tells the maximum number of times that the graph of a polynomial function intersects the x-axis.
• If the value of f(x) changes signs from one value of x to the next, then there is a zero between those two x-values.
• Extrema occur at relative maxima or minima of the function.

Lessons 2-3 and 2-4 Operations with Polynomials

• Polynomials can be added or subtracted by performing the operations indicated and combining like terms.
• To subtract a polynomial, add its additive inverse.
• Polynomials can be multiplied by using the Distributive Property to multiply each term in one polynomial by each term in the other.
• The set of polynomials is closed under the operations of addition, subtraction, and multiplication.
• To multiply two binomials, you can use a shortcut called the FOIL method.
• You can divide a polynomial by a polynomial with more than one term by using a process similar to long division of real numbers.
• Synthetic division is an alternate method used to divide a polynomial by a binomial of degree 1.

Lesson 2-5 Powers of Binomials

• Pascal's triangle is a triangle of numbers in which a row represents the coefficients of an expanded binomial (a + b)^n. Each row begins and ends with 1. Each coefficient can be found by adding the two coefficients above it in the previous row.
• You can also use the Binomial Theorem to expand a binomial. If n is a natural number, then (a + b)^n = C0a^n b^0 + C1a^{n-1}b^1 + C2a^{n-2}b^2 + C3a^{n-3}b^3 + … + C_na^0b^n.