Answer :
To simplify
[tex]$$
(2x - 3)\left(5x^4 - 7x^3 + 6x^2 - 9\right),
$$[/tex]
we start by distributing each term in the binomial to every term in the polynomial.
1. Distribute the first term, [tex]$2x$[/tex], across the polynomial:
[tex]\[
\begin{aligned}
2x \cdot 5x^4 &= 10x^5, \\
2x \cdot (-7x^3) &= -14x^4, \\
2x \cdot 6x^2 &= 12x^3, \\
2x \cdot (-9) &= -18x.
\end{aligned}
\][/tex]
2. Distribute the second term, [tex]$-3$[/tex], across the polynomial:
[tex]\[
\begin{aligned}
-3 \cdot 5x^4 &= -15x^4, \\
-3 \cdot (-7x^3) &= 21x^3, \\
-3 \cdot 6x^2 &= -18x^2, \\
-3 \cdot (-9) &= 27.
\end{aligned}
\][/tex]
3. Now, combine like terms:
- For [tex]$x^5$[/tex]: The only term is [tex]$10x^5$[/tex].
- For [tex]$x^4$[/tex]: Combine [tex]$-14x^4$[/tex] and [tex]$-15x^4$[/tex] to get
[tex]$$
-14x^4 - 15x^4 = -29x^4.
$$[/tex]
- For [tex]$x^3$[/tex]: Combine [tex]$12x^3$[/tex] and [tex]$21x^3$[/tex] to get
[tex]$$
12x^3 + 21x^3 = 33x^3.
$$[/tex]
- For [tex]$x^2$[/tex]: The only term is [tex]$-18x^2$[/tex].
- For [tex]$x$[/tex]: The only term is [tex]$-18x$[/tex].
- The constant term is [tex]$27$[/tex].
4. Thus, the final expanded expression is:
[tex]$$
10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27.
$$[/tex]
This is the simplified form of the given expression.
[tex]$$
(2x - 3)\left(5x^4 - 7x^3 + 6x^2 - 9\right),
$$[/tex]
we start by distributing each term in the binomial to every term in the polynomial.
1. Distribute the first term, [tex]$2x$[/tex], across the polynomial:
[tex]\[
\begin{aligned}
2x \cdot 5x^4 &= 10x^5, \\
2x \cdot (-7x^3) &= -14x^4, \\
2x \cdot 6x^2 &= 12x^3, \\
2x \cdot (-9) &= -18x.
\end{aligned}
\][/tex]
2. Distribute the second term, [tex]$-3$[/tex], across the polynomial:
[tex]\[
\begin{aligned}
-3 \cdot 5x^4 &= -15x^4, \\
-3 \cdot (-7x^3) &= 21x^3, \\
-3 \cdot 6x^2 &= -18x^2, \\
-3 \cdot (-9) &= 27.
\end{aligned}
\][/tex]
3. Now, combine like terms:
- For [tex]$x^5$[/tex]: The only term is [tex]$10x^5$[/tex].
- For [tex]$x^4$[/tex]: Combine [tex]$-14x^4$[/tex] and [tex]$-15x^4$[/tex] to get
[tex]$$
-14x^4 - 15x^4 = -29x^4.
$$[/tex]
- For [tex]$x^3$[/tex]: Combine [tex]$12x^3$[/tex] and [tex]$21x^3$[/tex] to get
[tex]$$
12x^3 + 21x^3 = 33x^3.
$$[/tex]
- For [tex]$x^2$[/tex]: The only term is [tex]$-18x^2$[/tex].
- For [tex]$x$[/tex]: The only term is [tex]$-18x$[/tex].
- The constant term is [tex]$27$[/tex].
4. Thus, the final expanded expression is:
[tex]$$
10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27.
$$[/tex]
This is the simplified form of the given expression.
