Question

(a - b)² = (a - b)(a - b) = a² - 2ab + b²

a. the product of (a + b) and (a - b) is the square of (a) minus the square of (b). ((a + b)(a - b) = (a - b)(a + b) = a² - b²)

1. ____ using the distributive property to factor polynomials with four or more terms.

1. ____ if the product of two factors is 0, then at least one of the factors must be 0.

1. ____ can be written in the standard form (ax² + bx + c = 0) where (a

eq 0)

1. ____ a polynomial that cannot be written as a product of two polynomials with integral coefficients.

1. ____ represents the subtraction of one perfect square from another. (a² - b² = (a + b)(a - b)) or ((a - b)(a + b))

1. ____ is a quadratic expression that can be written as the square of a binomial. it has the form: ((a + b)² = (a + b)(a + b) = a² + ab + ab + b² = a² + 2ab + b²); ((a - b)² = (a - b)(a - b) = a² - ab - ab + b² = a² - 2ab + b²)

1. ____ to solve a quadratic equation in the form (x² = n), take the square root of each side.

1. ____ for any number (n geq 0), if (x² = n), then (x = pm sqrt{n})

Explanation:

To solve these fill - in - the - blank questions related to algebraic concepts, we analyze each one:

Question 4

The Distributive Property is used to expand polynomials with four or more terms. When we have a polynomial with multiple terms and we want to factor it (or expand it), the Distributive Property (also known as the distributive law of multiplication over addition/subtraction) is employed. For example, when factoring \(ax+bx + ay+by\), we can use the distributive property as \(x(a + b)+y(a + b)=(a + b)(x + y)\). So the answer here is "expand".

Question 5

The Zero - Product Property states that if the product of two factors is 0, then at least one of the factors must be 0. Mathematically, if \(ab = 0\), then either \(a=0\) or \(b = 0\) (or both). So the answer is "Zero - Product Property".

Question 6

A quadratic equation can be written in the standard form \(ax^{2}+bx + c = 0\), where \(a
eq0\). This is the standard form of a quadratic equation, which is a second - degree polynomial equation in one variable. So the answer is "Quadratic Equation".

Question 7

A prime polynomial is a polynomial that cannot be written as a product of two polynomials with integral coefficients (other than 1 and itself). For example, \(x^{2}+x + 1\) is a prime polynomial over the integers because it cannot be factored into two polynomials with integer coefficients. So the answer is "Prime Polynomial".

Question 8

The difference of squares formula represents the subtraction of one perfect square from another. The formula for the difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a^{2}\) and \(b^{2}\) are perfect squares. So the answer is "Difference of Squares".

Question 9

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. For example, \(a^{2}+2ab + b^{2}=(a + b)^{2}\) and \(a^{2}-2ab + b^{2}=(a - b)^{2}\) are perfect square trinomials. So the answer is "Perfect Square Trinomial".

Question 10

To solve a quadratic equation in the form \(x^{2}=n\), we take the square root of each side. If \(x^{2}=n\) (\(n\geq0\)), then \(x=\pm\sqrt{n}\). So the operation we perform is "take the square root of".

Final Answers

1. expand
2. Zero - Product Property
3. Quadratic Equation
4. Prime Polynomial
5. Difference of Squares
6. Perfect Square Trinomial
7. take the square root of

Answer:

To solve these fill - in - the - blank questions related to algebraic concepts, we analyze each one:

Question 4

The Distributive Property is used to expand polynomials with four or more terms. When we have a polynomial with multiple terms and we want to factor it (or expand it), the Distributive Property (also known as the distributive law of multiplication over addition/subtraction) is employed. For example, when factoring \(ax+bx + ay+by\), we can use the distributive property as \(x(a + b)+y(a + b)=(a + b)(x + y)\). So the answer here is "expand".

Question 5

The Zero - Product Property states that if the product of two factors is 0, then at least one of the factors must be 0. Mathematically, if \(ab = 0\), then either \(a=0\) or \(b = 0\) (or both). So the answer is "Zero - Product Property".

Question 6

A quadratic equation can be written in the standard form \(ax^{2}+bx + c = 0\), where \(a
eq0\). This is the standard form of a quadratic equation, which is a second - degree polynomial equation in one variable. So the answer is "Quadratic Equation".

Question 7

A prime polynomial is a polynomial that cannot be written as a product of two polynomials with integral coefficients (other than 1 and itself). For example, \(x^{2}+x + 1\) is a prime polynomial over the integers because it cannot be factored into two polynomials with integer coefficients. So the answer is "Prime Polynomial".

Question 8

The difference of squares formula represents the subtraction of one perfect square from another. The formula for the difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a^{2}\) and \(b^{2}\) are perfect squares. So the answer is "Difference of Squares".

Question 9

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. For example, \(a^{2}+2ab + b^{2}=(a + b)^{2}\) and \(a^{2}-2ab + b^{2}=(a - b)^{2}\) are perfect square trinomials. So the answer is "Perfect Square Trinomial".

Question 10

To solve a quadratic equation in the form \(x^{2}=n\), we take the square root of each side. If \(x^{2}=n\) (\(n\geq0\)), then \(x=\pm\sqrt{n}\). So the operation we perform is "take the square root of".

Final Answers

1. expand
2. Zero - Product Property
3. Quadratic Equation
4. Prime Polynomial
5. Difference of Squares
6. Perfect Square Trinomial
7. take the square root of