Grade Nine Math Concepts Review (Algebra, Polynomials, Equations)

Flashcards covering exponent laws, polynomials, and solving equations as presented in the Grade 9 review video.

What does the product of powers rule state when multiplying a^m by a^n with the same base?

Keep the base and add the exponents: a^m * a^n = a^(m+n).

What does the quotient of powers rule state for the same base?

Keep the base and subtract the exponents: a^m / a^n = a^(m-n).

What is the power of a power rule?

(a^m)^n = a^(m·n).

How do you apply a power to a quotient: (a/b)^x?

Distribute the exponent to numerator and denominator: (a/b)^x = a^x / b^x.

How does the power to a product rule work: (ab)^x?

Distribute the exponent to each factor: (ab)^x = a^x b^x.

What is any nonzero base to the power of 0?

Equals 1 (a^0 = 1).

What is the negative exponent rule?

a^(-n) = 1 / a^n; rewrite with a positive exponent.

Compute 4^3 * 4^5.

Calculation: 4^3 \cdot 4^5### Explanation: This problem uses the product of powers rule, which states that when multiplying powers with the same base, you keep the base and add the exponents (a^m \cdot a^n = a^{m+n}).### Steps:1. Identify the base: The base is 4.2. Identify the exponents: The exponents are 3 and 5.3. Apply the rule: Add the exponents: 3 + 5 = 8.### Answer: 4^8

Compute 7^7 / 7^2.

Calculation: 7^7 / 7^2### Explanation: This problem uses the quotient of powers rule, which states that when dividing powers with the same base, you keep the base and subtract the exponents (a^m / a^n = a^{m-n}).### Steps:1. Identify the base: The base is 7.2. Identify the exponents: The exponents are 7 (numerator) and 2 (denominator).3. Apply the rule: Subtract the denominator's exponent from the numerator's exponent: 7 - 2 = 5.### Answer: 7^5

What is a term in a polynomial?

A product of numbers and variables, e.g., 2x or 3yz^2.

What is a polynomial?

An expression with one or more terms connected by addition or subtraction.

What is a monomial, a binomial, and a trinomial?

Monomial = one term; Binomial = two terms; Trinomial = three terms.

How do you determine the degree of a term like 3x^2y?

How to determine the degree of a term: To find the degree of a single term (like 3x^2y), you sum the exponents of all its variables.### Steps:1. Identify all variables in the term: In 3x^2y, the variables are x and y.2. Identify their exponents: The exponent for x is 2. The exponent for y is 1 (since y is equivalent to y^1).3. Sum the exponents: Add the exponents together: 2 + 1 = 3.### Answer: The degree of the term 3x^2y is 3.

How do you determine the degree of a polynomial?

The degree is the highest degree among its terms.

What are like terms?

Terms with identical variables raised to identical exponents (coefficients may differ).

How do you combine like terms?

Add or subtract coefficients while keeping the variable part the same.

What is the distributive property in simple terms?

Multiply a term outside the parentheses by each term inside: a(b+c) = ab + ac.

Give an example of distributive property with a monomial outside: 5(4x+2).

Problem: Apply the distributive property to 5(4x+2)### Explanation: The distributive property involves multiplying the term outside the parentheses by each term inside the parentheses. The rule is a(b+c) = ab + ac.### Steps:1. Multiply the outside term (5) by the first term inside (4x): 5 \cdot 4x = 20x.2. Multiply the outside term (5) by the second term inside (2): 5 \cdot 2 = 10.3. Combine the results with the original operation between the terms: 20x + 10.### Answer: 5(4x+2) = 20x + 10

How do you apply distributive property to a monomial times a trinomial?

Multiply the outside term by each term inside the brackets, then combine like terms if possible.

What is the method to isolate a variable using the balance method?

Move other terms to the opposite side using opposite operations so the variable is by itself.

Solve x + 4 = 7.

Equation: x + 4 = 7### Explanation: To solve a linear equation, use the balance method to isolate the variable. Perform the inverse operation on both sides of the equation to maintain equality.### Steps:1. Identify the operation affecting x: Addition of 4.2. Perform the inverse operation on both sides: Subtract 4 from both sides. x + 4 - 4 = 7 - 43. Simplify both sides: x = 3### Answer: x = 3### Check: Substitute x=3 back into the original equation: 3 + 4 = 7, which is true.

Solve g - 5 = -3.

Equation: g - 5 = -3### Explanation: To solve a linear equation, use the balance method to isolate the variable. Perform the inverse operation on both sides of the equation to maintain equality.### Steps:1. Identify the operation affecting g: Subtraction of 5.2. Perform the inverse operation on both sides: Add 5 to both sides. g - 5 + 5 = -3 + 53. Simplify both sides: g = 2### Answer: g = 2### Check: Substitute g=2 back into the original equation: 2 - 5 = -3, which is true.

Solve 5u = -20.

Equation: 5u = -20### Explanation: To solve a linear equation, use the balance method to isolate the variable. Perform the inverse operation on both sides of the equation to maintain equality.### Steps:1. Identify the operation affecting u: Multiplication by 5.2. Perform the inverse operation on both sides: Divide both sides by 5. 5u / 5 = -20 / 53. Simplify both sides: u = -4### Answer: u = -4### Check: Substitute u=-4 back into the original equation: 5(-4) = -20, which is true.

Solve 7y + 8 = 15.

Equation: 7y + 8 = 15### Explanation: This is a two-step equation. First, isolate the term with the variable by adding or subtracting constants. Then, isolate the variable by dividing or multiplying.### Steps:1. Subtract 8 from both sides (to isolate the 7y term): 7y + 8 - 8 = 15 - 8 7y = 72. Divide both sides by 7 (to isolate y): 7y / 7 = 7 / 7 y = 1### Answer: y = 1### Check: Substitute y=1 back into the original equation: 7(1) + 8 = 7 + 8 = 15, which is true.

How do you solve a two-step equation like 7y + 8 = 15?

Subtract 8 to get 7y = 7, then divide by 7 to get y = 1.

What is cross multiplication and when should you use it?

For equations of the form a/b = c/d; cross-multiply: ad = bc; only valid for fraction equals fraction.

How do you clear fractions by using a common denominator?

Multiply both sides by the least common multiple of the denominators to eliminate fractions.

Solve 6x - 24 = 5x - 15.

Equation: 6x - 24 = 5x - 15### Explanation: To solve an equation with variables on both sides, first gather all variable terms on one side and all constant terms on the other side. Then, solve the resulting two-step equation.### Steps:1. Subtract 5x from both sides (to gather variable terms on the left): 6x - 5x - 24 = 5x - 5x - 15 x - 24 = -152. Add 24 to both sides (to gather constant terms on the right): x - 24 + 24 = -15 + 24 x = 9### Answer: x = 9### Check: Substitute x=9 into the left side: 6(9) - 24 = 54 - 24 = 30. Substitute x=9 into the right side: 5(9) - 15 = 45 - 15 = 30. Since 30 = 30, the solution is correct.

Word problem: Natasha has 250 more than Kristen; together they have 880. How many does each have?

Problem: Natasha has 250 more than Kristen; together they have 880. How many does each have?### Explanation: This is a system of equations that can be solved by setting up variables for each person's amount and then using substitution or elimination.### Steps:1. Define variables: Let K be the amount Kristen has, and N be the amount Natasha has.2. Set up equations based on the problem:

• Equation 1 (Natasha has 250 more than Kristen): N = K + 250
• Equation 2 (Together they have 880): K + N = 8803. Substitute Equation 1 into Equation 2:
  K + (K + 250) = 8804. Simplify and solve for K:
  2K + 250 = 880
  2K = 880 - 250
  2K = 630
  K = 630 / 2
  K = 3155. Substitute the value of K back into Equation 1 to find N:
  N = 315 + 250
  N = 565### Answer: Kristen has 315. Natasha has 565.### Check: 315 + 565 = 880. Natasha (565) has 250 more than Kristen (315), since 565 - 315 = 250. The solution is correct.

What is a constant term?

A term with no variable; just a number.

In what order should polynomial terms be written by degree?

Descending order: highest degree term first.