Answer :
To simplify the product
[tex]$$\left(4x^3 - 2x + 7\right)\left(2x^2 - 6x - 3\right),$$[/tex]
we will use the distributive property by multiplying each term in the first polynomial by every term in the second polynomial. Follow these steps:
1. Multiply the first term of the first polynomial, [tex]$4x^3$[/tex], by every term in the second polynomial:
[tex]\[
\begin{aligned}
4x^3 \cdot 2x^2 &= 8x^5, \\
4x^3 \cdot (-6x) &= -24x^4, \\
4x^3 \cdot (-3) &= -12x^3.
\end{aligned}
\][/tex]
2. Multiply the second term of the first polynomial, [tex]$-2x$[/tex], by every term in the second polynomial:
[tex]\[
\begin{aligned}
-2x \cdot 2x^2 &= -4x^3, \\
-2x \cdot (-6x) &= 12x^2, \\
-2x \cdot (-3) &= 6x.
\end{aligned}
\][/tex]
3. Multiply the third term of the first polynomial, [tex]$7$[/tex], by every term in the second polynomial:
[tex]\[
\begin{aligned}
7 \cdot 2x^2 &= 14x^2, \\
7 \cdot (-6x) &= -42x, \\
7 \cdot (-3) &= -21.
\end{aligned}
\][/tex]
4. Now, combine like terms:
- The [tex]$x^5$[/tex] term:
[tex]\[
8x^5.
\][/tex]
- The [tex]$x^4$[/tex] term:
[tex]\[
-24x^4.
\][/tex]
- The [tex]$x^3$[/tex] terms:
[tex]\[
-12x^3 - 4x^3 = -16x^3.
\][/tex]
- The [tex]$x^2$[/tex] terms:
[tex]\[
12x^2 + 14x^2 = 26x^2.
\][/tex]
- The [tex]$x$[/tex] terms:
[tex]\[
6x - 42x = -36x.
\][/tex]
- The constant term:
[tex]\[
-21.
\][/tex]
5. Hence, the fully expanded and simplified expression is:
[tex]\[
8x^5 - 24x^4 - 16x^3 + 26x^2 - 36x - 21.
\][/tex]
Thus, the final answer is:
[tex]$$\boxed{8x^5 - 24x^4 - 16x^3 + 26x^2 - 36x - 21}.$$[/tex]
[tex]$$\left(4x^3 - 2x + 7\right)\left(2x^2 - 6x - 3\right),$$[/tex]
we will use the distributive property by multiplying each term in the first polynomial by every term in the second polynomial. Follow these steps:
1. Multiply the first term of the first polynomial, [tex]$4x^3$[/tex], by every term in the second polynomial:
[tex]\[
\begin{aligned}
4x^3 \cdot 2x^2 &= 8x^5, \\
4x^3 \cdot (-6x) &= -24x^4, \\
4x^3 \cdot (-3) &= -12x^3.
\end{aligned}
\][/tex]
2. Multiply the second term of the first polynomial, [tex]$-2x$[/tex], by every term in the second polynomial:
[tex]\[
\begin{aligned}
-2x \cdot 2x^2 &= -4x^3, \\
-2x \cdot (-6x) &= 12x^2, \\
-2x \cdot (-3) &= 6x.
\end{aligned}
\][/tex]
3. Multiply the third term of the first polynomial, [tex]$7$[/tex], by every term in the second polynomial:
[tex]\[
\begin{aligned}
7 \cdot 2x^2 &= 14x^2, \\
7 \cdot (-6x) &= -42x, \\
7 \cdot (-3) &= -21.
\end{aligned}
\][/tex]
4. Now, combine like terms:
- The [tex]$x^5$[/tex] term:
[tex]\[
8x^5.
\][/tex]
- The [tex]$x^4$[/tex] term:
[tex]\[
-24x^4.
\][/tex]
- The [tex]$x^3$[/tex] terms:
[tex]\[
-12x^3 - 4x^3 = -16x^3.
\][/tex]
- The [tex]$x^2$[/tex] terms:
[tex]\[
12x^2 + 14x^2 = 26x^2.
\][/tex]
- The [tex]$x$[/tex] terms:
[tex]\[
6x - 42x = -36x.
\][/tex]
- The constant term:
[tex]\[
-21.
\][/tex]
5. Hence, the fully expanded and simplified expression is:
[tex]\[
8x^5 - 24x^4 - 16x^3 + 26x^2 - 36x - 21.
\][/tex]
Thus, the final answer is:
[tex]$$\boxed{8x^5 - 24x^4 - 16x^3 + 26x^2 - 36x - 21}.$$[/tex]
