Answer :
To multiply the trinomials
[tex]$$\left(x^2 - x + 9\right)\left(x^2 + x + 15\right),$$[/tex]
we can use the distributive property (also known as the FOIL method for trinomials by distributing each term).
First, multiply each term in the first trinomial by each term in the second trinomial:
1. Multiply [tex]\( x^2 \)[/tex] by each term in the second trinomial:
[tex]$$
x^2 \cdot x^2 = x^4, \quad x^2 \cdot x = x^3, \quad x^2 \cdot 15 = 15x^2.
$$[/tex]
2. Multiply [tex]\( -x \)[/tex] by each term in the second trinomial:
[tex]$$
-x \cdot x^2 = -x^3, \quad -x \cdot x = -x^2, \quad -x \cdot 15 = -15x.
$$[/tex]
3. Multiply [tex]\( 9 \)[/tex] by each term in the second trinomial:
[tex]$$
9 \cdot x^2 = 9x^2, \quad 9 \cdot x = 9x, \quad 9 \cdot 15 = 135.
$$[/tex]
Now, list all the resulting terms:
[tex]$$
x^4 + x^3 + 15x^2 - x^3 - x^2 - 15x + 9x^2 + 9x + 135.
$$[/tex]
Next, combine like terms:
- The [tex]\( x^4 \)[/tex] term:
[tex]$$
x^4.
$$[/tex]
- The [tex]\( x^3 \)[/tex] terms:
[tex]$$
x^3 - x^3 = 0.
$$[/tex]
- The [tex]\( x^2 \)[/tex] terms:
[tex]$$
15x^2 - x^2 + 9x^2 = (15 - 1 + 9)x^2 = 23x^2.
$$[/tex]
- The [tex]\( x \)[/tex] terms:
[tex]$$
-15x + 9x = -6x.
$$[/tex]
- The constant term:
[tex]$$
135.
$$[/tex]
Thus, the product is:
[tex]$$
x^4 + 23x^2 - 6x + 135.
$$[/tex]
Comparing with the given options:
A. [tex]\( x^4 - 23x^2 - 16x + 135 \)[/tex]
B. [tex]\( x^4 + 23x^2 - 6x + 135 \)[/tex]
C. [tex]\( x^4 + 3x^2 - 16x + 135 \)[/tex]
D. [tex]\( x^4 - 3x^2 - 6x + 135 \)[/tex]
The expression [tex]\( x^4 + 23x^2 - 6x + 135 \)[/tex] corresponds to option B.
[tex]$$\left(x^2 - x + 9\right)\left(x^2 + x + 15\right),$$[/tex]
we can use the distributive property (also known as the FOIL method for trinomials by distributing each term).
First, multiply each term in the first trinomial by each term in the second trinomial:
1. Multiply [tex]\( x^2 \)[/tex] by each term in the second trinomial:
[tex]$$
x^2 \cdot x^2 = x^4, \quad x^2 \cdot x = x^3, \quad x^2 \cdot 15 = 15x^2.
$$[/tex]
2. Multiply [tex]\( -x \)[/tex] by each term in the second trinomial:
[tex]$$
-x \cdot x^2 = -x^3, \quad -x \cdot x = -x^2, \quad -x \cdot 15 = -15x.
$$[/tex]
3. Multiply [tex]\( 9 \)[/tex] by each term in the second trinomial:
[tex]$$
9 \cdot x^2 = 9x^2, \quad 9 \cdot x = 9x, \quad 9 \cdot 15 = 135.
$$[/tex]
Now, list all the resulting terms:
[tex]$$
x^4 + x^3 + 15x^2 - x^3 - x^2 - 15x + 9x^2 + 9x + 135.
$$[/tex]
Next, combine like terms:
- The [tex]\( x^4 \)[/tex] term:
[tex]$$
x^4.
$$[/tex]
- The [tex]\( x^3 \)[/tex] terms:
[tex]$$
x^3 - x^3 = 0.
$$[/tex]
- The [tex]\( x^2 \)[/tex] terms:
[tex]$$
15x^2 - x^2 + 9x^2 = (15 - 1 + 9)x^2 = 23x^2.
$$[/tex]
- The [tex]\( x \)[/tex] terms:
[tex]$$
-15x + 9x = -6x.
$$[/tex]
- The constant term:
[tex]$$
135.
$$[/tex]
Thus, the product is:
[tex]$$
x^4 + 23x^2 - 6x + 135.
$$[/tex]
Comparing with the given options:
A. [tex]\( x^4 - 23x^2 - 16x + 135 \)[/tex]
B. [tex]\( x^4 + 23x^2 - 6x + 135 \)[/tex]
C. [tex]\( x^4 + 3x^2 - 16x + 135 \)[/tex]
D. [tex]\( x^4 - 3x^2 - 6x + 135 \)[/tex]
The expression [tex]\( x^4 + 23x^2 - 6x + 135 \)[/tex] corresponds to option B.
