Polynomial Functions and Operations

Monomials

  • Definition: A monomial is a number, variable, or a product of numbers and variables.

  • Examples of Monomials:

    • 3,x,4y,5xyz2,7x3y2z3, x, 4y, 5xyz^2, 7x^3y^2z

  • Exponent Rules for Simplifying Monomial Expressions:

    • Product Rule: For any numbers or variables aa and bb, when multiplying:
      am×an=am+na^m \times a^n = a^{m+n}

    • Power Rule: For any number aa, when raising to a power:
      (am)n=amn(a^m)^n = a^{mn}

    • Quotient Rule: For any nonzero aa:
      aman=am−n\frac{a^m}{a^n} = a^{m-n}

    • Negative Exponent Rule: For any nonzero aa:
      a−n=1ana^{-n} = \frac{1}{a^n}

    • Zero Exponent Rule: For any nonzero aa:
      a0=1a^0 = 1

  • Combining Like Terms: When adding or subtracting monomials, one must combine like terms.

Examples of Combining Like Terms:

  1. 2x2+6x−5x2+7x=−3x2+13x2x^2 + 6x - 5x^2 + 7x = -3x^2 + 13x

  2. 3a2−9ab+(2)8−ab=3a2−9ab+163a^2 - 9ab + (2)8 - a^b = 3a^2 - 9ab + 16

  3. 6a4b4−4a4b3+5=6a4b4−4a4b3+56a^4b^4 - 4a^4b^3 + 5 = 6a^4b^4 - 4a^4b^3 + 5

  4. 4k2−7k+10−2k2=2k2−7k+104k^2 - 7k + 10 - 2k^2 = 2k^2 - 7k + 10

  5. 2r2+1/2(72)−3z2r^2 + 1/2(7^2) - 3z

  6. 12−3w−1412 - 3w - 14

  7. 15x4+x−7x=15x4−6x15x^4 + x - 7x = 15x^4 - 6x

  8. 4(c4)−4(c)=4c4−4c4(c^4) - 4(c) = 4c^4 - 4c

Example of Quotients:

  1. Provide an example of two monomials with a quotient of 2n3m\frac{2n}{3m}.

Polynomials

  • Definition: A polynomial is defined as the sum or difference of many monomials.

  • Highest Exponent: The highest exponent of a polynomial is called the degree.

  • Standard Form: A polynomial is in standard form when its terms are written in decreasing order of degree.

Writing Polynomials in Standard Form:

  1. 5k4−3k3+2k−8→5k4−3k3+2k−85k^4 - 3k^3 + 2k - 8 \rightarrow 5k^4 - 3k^3 + 2k - 8

  2. 2a2b−3b+b2+2a→2a2b+b2+2a−3b2a^2b - 3b + b^2 + 2a \rightarrow 2a^2b + b^2 + 2a - 3b

  3. 9x2+2xy−6xy2→−6xy2+9x2+2xy9x^2 + 2xy - 6xy^2 \rightarrow -6xy^2 + 9x^2 + 2xy

Classifying Polynomials

  • Classify polynomials by degree (highest exponent) and by the number of terms:

    • Number of Terms: 1 (monomial), 2 (binomial), 3 (trinomial), 4+ (polynomial)

    • Degree: 0 (constant), 1 (linear), 2 (quadratic), 3 (cubic), 4 (quartic), 5 (quintic) …

Examples of Classification:

  1. 3x+13x + 1 (Linear binomial, Degree 1)

  2. 5x4−9x2+25x^4 - 9x^2 + 2 (Quartic trinomial, Degree 4)

  3. 2424 (Constant monomial, Degree 0)

  4. 2x3+3x2+4x+52x^3 + 3x^2 + 4x + 5 (Cubic polynomial, Degree 3)

  5. x2−2x+2x^2 - 2x + 2 (Quadratic trinomial, Degree 2)

  6. 33 (Constant monomial, Degree 0)

  7. Example of cubic binomial: x3−2x^3 - 2

  8. Example of linear monomial: 2a2a

Adding & Subtracting Polynomials

  • Combine Like Terms: Remember to watch out for subtraction problems!

  • Answer must be in standard form

Examples:

  1. (2x2+3x2)+(11x−5x2−4)=5x2+11x−5x2−4=11x−4(2x^2 + 3x^2) + (11x - 5x^2 - 4) = 5x^2 + 11x - 5x^2 - 4 = 11x - 4

  2. 9n2−4(2n+10)−2−(3n+7)=9n2−8n−40−2−3n−7=9n2−11n−499n^2 - 4(2n + 10) - 2 - (3n + 7) = 9n^2 - 8n - 40 - 2 - 3n - 7 = 9n^2 - 11n - 49

  3. Implement the usual conventions for combining like terms when performing polynomial operations.

  4. Subtract −3ab-3ab from 2=2−(−3ab)=2+3ab2 = 2 - (-3ab) = 2 + 3ab

Multiplying Polynomials

  • Distribute or Use FOIL (First, Outside, Inside, Last)

Example Multiplications:

  1. (2w2−7)(w2+1)=2w4+2w2−7w2−7=2w4−5w2−7(2w^2 - 7)(w^2 + 1) = 2w^4 + 2w^2 - 7w^2 - 7 = 2w^4 - 5w^2 - 7

  2. (x+1)(3x+7)=3x2+7x+3x+7=3x2+10x+7(x + 1)(3x + 7) = 3x^2 + 7x + 3x + 7 = 3x^2 + 10x + 7

  3. 3(a+4)(2a+3)=3(2a2+3a+8a+12)=3(2a2+11a+12)=6a2+33a+363(a + 4)(2a + 3) = 3(2a^2 + 3a + 8a + 12) = 3(2a^2 + 11a + 12) = 6a^2 + 33a + 36

  4. (2+3)(5+6)=(5)(11)=55(2 + 3)(5 + 6) = (5)(11) = 55

Dividing Polynomials

  • To divide each term in the numerator by the monomial in the denominator, rewrite all terms with negative exponents and write the answer in standard form.

Examples:

  1. 9m324m3=38\frac{9m^3}{24m^3} = \frac{3}{8}

  2. 20x4−8x3+4x2+168x4=20x48x4−8x38x4+4x28x4+168x4=52−x−1+12x−2+2x−4\frac{20x^4 - 8x^3 + 4x^2 + 16}{8x^4} = \frac{20x^4}{8x^4} - \frac{8x^3}{8x^4} + \frac{4x^2}{8x^4} + \frac{16}{8x^4} = \frac{5}{2} - x^{-1} + \frac{1}{2}x^{-2} + 2x^{-4}

  3. 14x2+10a35b2x2=14x235b2x2+10a35b2x2=25b2+2a7b2x2\frac{14x^2 + 10a}{35b^2x^2} = \frac{14x^2}{35b^2x^2} + \frac{10a}{35b^2x^2} = \frac{2}{5b^2} + \frac{2a}{7b^2x^2}

Polynomial Functions

  • A polynomial function is defined as a function of the form:

    f(x)=a<em>nxn+a</em>n−1xn−1+⋯+a<em>0f(x) = a<em>nx^n + a</em>{n-1}x^{n-1} + \dots + a<em>0 where a</em>n,…,a0a</em>n, \dots, a_0 are real numbers, nn is a positive integer (and no negative exponents).

  • The leading coefficient of a polynomial is the coefficient of the term with the highest degree.

Classifying Polynomial Functions by Degree:

  • Degree (e.g.: f(x)=4f(x) = 4 is a constant; f(x)=12x+3f(x)= 12x + 3 is a linear function; etc.)

  • Leading Coefficient examples:

    • For f(x)=5x3−2x+7f(x) = 5x^3 - 2x + 7, the leading coefficient is 55.

    • For g(x)=−x2+4xg(x) = -x^2 + 4x, the leading coefficient is −1-1.

Polynomial Parent Functions

  • Identify and sketch parent functions: constant, linear, quadratic, cubic, quartic, quintic, etc..

  • End Behaviors are determined by the leading coefficient and the degree of the polynomial (even or odd).

Examples Determining End Behavior:

  1. Even Degree (e.g., f(x)=x2,f(x)=x4f(x) = x^2, f(x) = x^4):

    • Positive Leading Coefficient (a_n > 0): Both ends go up.

      • As x→−∞x \rightarrow -\infty, f(x)→∞f(x) \rightarrow \infty

      • As x→∞x \rightarrow \infty, f(x)→∞f(x) \rightarrow \infty

    • Negative Leading Coefficient (a_n < 0): Both ends go down.

      • As x→−∞x \rightarrow -\infty, f(x)→−∞f(x) \rightarrow -\infty

      • As x→∞x \rightarrow \infty, f(x)→−∞f(x) \rightarrow -\infty

  2. Odd Degree (e.g., f(x)=x3,f(x)=x5f(x) = x^3, f(x) = x^5):

    • Positive Leading Coefficient (a_n > 0): Left end down, right end up.

      • As x→−∞x \rightarrow -\infty, f(x)→−∞f(x) \rightarrow -\infty

      • As x→∞x \rightarrow \infty, f(x)→∞f(x) \rightarrow \infty

    • Negative Leading Coefficient (a_n < 0): Left end up, right end down.

      • As x→−∞x \rightarrow -\infty, f(x)→∞f(x) \rightarrow \infty

      • As x→∞x \rightarrow \infty, f(x)→−∞f(x) \rightarrow -\infty