Matching Polynomial Functions with their Leading Coefficients

Oct 13, 2025 by windailys.com 13 views

What is the Value of $6(x-2)(x-3) + 4(x-2)(x-5)$ when $x = -3$ ?

To evaluate the given expression, we need to substitute $x = -3$ into the expression and simplify.

First, let's expand the expression:

$6(x-2)(x-3) + 4(x-2)(x-5)$

Using the distributive property, we get:

$6(x^2 - 5x + 6) + 4(x^2 - 7x + 10)$

Expanding further, we get:

$6x^2 - 30x + 36 + 4x^2 - 28x + 40$

Combining like terms, we get:

$10x^2 - 58x + 76$

Now, let's substitute $x = -3$ into the expression:

$10(-3)^2 - 58(-3) + 76$

Simplifying, we get:

$10(9) + 174 + 76$

$= 90 + 174 + 76$

$= 340$

Therefore, the value of the expression when $x = -3$ is 340.

Matching Polynomial Functions with their Leading Coefficients

A. $P(x) = (x+2)(2x-3)(4x+7)$

To find the leading coefficient of the polynomial function, we need to multiply the three binomials:

$(x+2)(2x-3)(4x+7)$

Using the distributive property, we get:

$(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42$

Combining like terms, we get:

$8x^3 + 44x^2 + 11x - 42$

The leading coefficient of the polynomial function is 8.

B. $P(x) = \frac{1}{2}(x-2)(2x-3)(4x+7)$

To find the leading coefficient of the polynomial function, we need to multiply the three binomials:

$\frac{1}{2}(x-2)(2x-3)(4x+7)$

Using the distributive property, we get:

$\frac{1}{2}(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$\frac{1}{2}(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$\frac{1}{2}(8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42)$

Combining like terms, we get:

$\frac{1}{2}(8x^3 + 44x^2 + 11x - 42)$

The leading coefficient of the polynomial function is $\frac{1}{2}(8) = 4$ .

C. $P(x) = (x+2)(2x-3)(4x+7)$

To find the leading coefficient of the polynomial function, we need to multiply the three binomials:

$(x+2)(2x-3)(4x+7)$

Using the distributive property, we get:

$(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42$

Combining like terms, we get:

$8x^3 + 44x^2 + 11x - 42$

The leading coefficient of the polynomial function is 8.

D. $P(x) = (x+2)(2x-3)(4x+7)$

To find the leading coefficient of the polynomial function, we need to multiply the three binomials:

$(x+2)(2x-3)(4x+7)$

Using the distributive property, we get:

$(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42$

Combining like terms, we get:

$8x^3 + 44x^2 + 11x - 42$

The leading coefficient of the polynomial function is 8.

Matching Polynomial Functions with their Leading Coefficients

Polynomial Function	Leading Coefficient
A. $P(x) = (x+2)(2x-3)(4x+7)$	8
B. $P(x) = \frac{1}{2}(x-2)(2x-3)(4x+7)$	4
C. $P(x) = (x+2)(2x-3)(4x+7)$	8
D. $P(x) = (x+2)(2x-3)(4x+7)$	8

Therefore, the correct matching of polynomial functions with their leading coefficients is:

A. $P(x) = (x+2)(2x-3)(4x+7)$ has a leading coefficient of 8.
B. $P(x) = \frac{1}{2}(x-2)(2x-3)(4x+7)$ has a leading coefficient of 4.
C. $P(x) = (x+2)(2x-3)(4x+7)$ has a leading coefficient of 8.
D. $P(x) = (x+2)(2x-3)(4x+7)$ has a leading coefficient of 8.
Evaluating Algebraic Expressions and Matching Polynomial Functions: Q&A =====================================================================

Q: What is the value of $6(x-2)(x-3) + 4(x-2)(x-5)$ when $x = -3$ ?

A: To evaluate the given expression, we need to substitute $x = -3$ into the expression and simplify. First, let's expand the expression:

$6(x-2)(x-3) + 4(x-2)(x-5)$

Using the distributive property, we get:

$6(x^2 - 5x + 6) + 4(x^2 - 7x + 10)$

Expanding further, we get:

$6x^2 - 30x + 36 + 4x^2 - 28x + 40$

Combining like terms, we get:

$10x^2 - 58x + 76$

Now, let's substitute $x = -3$ into the expression:

$10(-3)^2 - 58(-3) + 76$

Simplifying, we get:

$10(9) + 174 + 76$

$= 90 + 174 + 76$

$= 340$

Therefore, the value of the expression when $x = -3$ is 340.

Q: How do I find the leading coefficient of a polynomial function?

A: To find the leading coefficient of a polynomial function, you need to multiply the three binomials:

$(x+2)(2x-3)(4x+7)$

Using the distributive property, we get:

$(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42$

Combining like terms, we get:

$8x^3 + 44x^2 + 11x - 42$

The leading coefficient of the polynomial function is 8.

Q: What is the difference between the leading coefficient and the constant term of a polynomial function?

A: The leading coefficient of a polynomial function is the coefficient of the highest degree term, while the constant term is the term that does not have a variable. For example, in the polynomial function $8x^3 + 44x^2 + 11x - 42$ , the leading coefficient is 8 and the constant term is -42.

Q: How do I match polynomial functions with their leading coefficients?

A: To match polynomial functions with their leading coefficients, you need to multiply the three binomials and combine like terms. Then, identify the leading coefficient of the resulting polynomial function.

Q: What is the importance of identifying the leading coefficient of a polynomial function?

A: Identifying the leading coefficient of a polynomial function is important because it helps you understand the behavior of the function. The leading coefficient determines the direction and rate of change of the function.

Q: Can you provide an example of a polynomial function with a leading coefficient of 4?

A: Yes, here is an example of a polynomial function with a leading coefficient of 4:

$P(x) = \frac{1}{2}(x-2)(2x-3)(4x+7)$

Using the distributive property, we get:

$\frac{1}{2}(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$\frac{1}{2}(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$\frac{1}{2}(8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42)$

Combining like terms, we get:

$\frac{1}{2}(8x^3 + 44x^2 + 11x - 42)$

The leading coefficient of the polynomial function is $\frac{1}{2}(8) = 4$ .

Q: Can you provide an example of a polynomial function with a leading coefficient of 8?

A: Yes, here is an example of a polynomial function with a leading coefficient of 8:

$P(x) = (x+2)(2x-3)(4x+7)$

Using the distributive property, we get:

$(2x^2 - 3x + 8x - 6)(4x + 7)$

Expanding further, we get:

$(2x^2 + 5x - 6)(4x + 7)$

Multiplying again, we get:

$8x^3 + 14x^2 + 20x^2 + 35x - 24x - 42$

Combining like terms, we get:

$8x^3 + 44x^2 + 11x - 42$

The leading coefficient of the polynomial function is 8.