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Answer :
To factor the polynomial
[tex]$$7x^5 + 21x^4 - 28x^3,$$[/tex]
we start by identifying the greatest common factor (GCF) of all the terms.
Step 1. Identify the GCF
Each term in the polynomial has a factor of [tex]$7$[/tex] and at least [tex]$x^3$[/tex]. Thus, the GCF is
[tex]$$7x^3.$$[/tex]
Step 2. Factor out the GCF
Divide every term by [tex]$7x^3$[/tex]:
[tex]\[
\begin{aligned}
7x^5 & = 7x^3 \cdot x^2, \\
21x^4 & = 7x^3 \cdot 3x, \\
-28x^3 & = 7x^3 \cdot (-4).
\end{aligned}
\][/tex]
So, the polynomial becomes:
[tex]$$7x^5 + 21x^4 - 28x^3 = 7x^3 \left(x^2 + 3x - 4\right).$$[/tex]
Step 3. Factor the quadratic
Now, we factor the quadratic expression
[tex]$$x^2 + 3x - 4.$$[/tex]
We need two numbers that multiply to [tex]$-4$[/tex] (the constant term) and add to [tex]$3$[/tex] (the coefficient of [tex]$x$[/tex]). The numbers [tex]$4$[/tex] and [tex]$-1$[/tex] work because:
[tex]\[
4 \times (-1) = -4 \quad \text{and} \quad 4 + (-1) = 3.
\][/tex]
Thus, the quadratic factors as:
[tex]$$x^2 + 3x - 4 = (x - 1)(x + 4).$$[/tex]
Step 4. Write the final factorization
Substitute the factorized quadratic back into the expression:
[tex]$$7x^5 + 21x^4 - 28x^3 = 7x^3 (x - 1)(x + 4).$$[/tex]
This is the complete factorization of the given polynomial.
[tex]$$7x^5 + 21x^4 - 28x^3,$$[/tex]
we start by identifying the greatest common factor (GCF) of all the terms.
Step 1. Identify the GCF
Each term in the polynomial has a factor of [tex]$7$[/tex] and at least [tex]$x^3$[/tex]. Thus, the GCF is
[tex]$$7x^3.$$[/tex]
Step 2. Factor out the GCF
Divide every term by [tex]$7x^3$[/tex]:
[tex]\[
\begin{aligned}
7x^5 & = 7x^3 \cdot x^2, \\
21x^4 & = 7x^3 \cdot 3x, \\
-28x^3 & = 7x^3 \cdot (-4).
\end{aligned}
\][/tex]
So, the polynomial becomes:
[tex]$$7x^5 + 21x^4 - 28x^3 = 7x^3 \left(x^2 + 3x - 4\right).$$[/tex]
Step 3. Factor the quadratic
Now, we factor the quadratic expression
[tex]$$x^2 + 3x - 4.$$[/tex]
We need two numbers that multiply to [tex]$-4$[/tex] (the constant term) and add to [tex]$3$[/tex] (the coefficient of [tex]$x$[/tex]). The numbers [tex]$4$[/tex] and [tex]$-1$[/tex] work because:
[tex]\[
4 \times (-1) = -4 \quad \text{and} \quad 4 + (-1) = 3.
\][/tex]
Thus, the quadratic factors as:
[tex]$$x^2 + 3x - 4 = (x - 1)(x + 4).$$[/tex]
Step 4. Write the final factorization
Substitute the factorized quadratic back into the expression:
[tex]$$7x^5 + 21x^4 - 28x^3 = 7x^3 (x - 1)(x + 4).$$[/tex]
This is the complete factorization of the given polynomial.
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