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Answer :
Sure! Let me show you how to factor the polynomial [tex]\( x^6 - 7x^4 + 6x^2 \)[/tex] step-by-step.
1. Identify Common Factors:
Notice that each term in the polynomial has a factor of [tex]\( x^2 \)[/tex]. So, we can factor [tex]\( x^2 \)[/tex] out of the whole expression:
[tex]\[
x^6 - 7x^4 + 6x^2 = x^2(x^4 - 7x^2 + 6)
\][/tex]
2. Factor the Quadratic Expression:
Now, we'll focus on factoring the quadratic expression inside the parentheses: [tex]\( x^4 - 7x^2 + 6 \)[/tex]. To make things simpler, let's substitute [tex]\( y = x^2 \)[/tex], turning the quadratic into:
[tex]\[
y^2 - 7y + 6
\][/tex]
3. Factor the Quadratic [tex]\( y^2 - 7y + 6 \)[/tex]:
We need two numbers that multiply to 6 (the constant term) and add to -7 (the linear coefficient). These numbers are -1 and -6.
So, we can factor it as:
[tex]\[
y^2 - 7y + 6 = (y - 1)(y - 6)
\][/tex]
4. Substitute Back:
Don't forget to substitute back [tex]\( y = x^2 \)[/tex]:
[tex]\[
(x^2 - 1)(x^2 - 6)
\][/tex]
5. Factor Further if Possible:
Now, look at each factor:
- [tex]\( x^2 - 1 \)[/tex] is a difference of squares and can be factored as [tex]\( (x - 1)(x + 1) \)[/tex].
So, the fully factored form of the polynomial is:
[tex]\[
x^2(x - 1)(x + 1)(x^2 - 6)
\][/tex]
And there we have it! The polynomial [tex]\( x^6 - 7x^4 + 6x^2 \)[/tex] is factored as [tex]\( x^2(x - 1)(x + 1)(x^2 - 6) \)[/tex].
1. Identify Common Factors:
Notice that each term in the polynomial has a factor of [tex]\( x^2 \)[/tex]. So, we can factor [tex]\( x^2 \)[/tex] out of the whole expression:
[tex]\[
x^6 - 7x^4 + 6x^2 = x^2(x^4 - 7x^2 + 6)
\][/tex]
2. Factor the Quadratic Expression:
Now, we'll focus on factoring the quadratic expression inside the parentheses: [tex]\( x^4 - 7x^2 + 6 \)[/tex]. To make things simpler, let's substitute [tex]\( y = x^2 \)[/tex], turning the quadratic into:
[tex]\[
y^2 - 7y + 6
\][/tex]
3. Factor the Quadratic [tex]\( y^2 - 7y + 6 \)[/tex]:
We need two numbers that multiply to 6 (the constant term) and add to -7 (the linear coefficient). These numbers are -1 and -6.
So, we can factor it as:
[tex]\[
y^2 - 7y + 6 = (y - 1)(y - 6)
\][/tex]
4. Substitute Back:
Don't forget to substitute back [tex]\( y = x^2 \)[/tex]:
[tex]\[
(x^2 - 1)(x^2 - 6)
\][/tex]
5. Factor Further if Possible:
Now, look at each factor:
- [tex]\( x^2 - 1 \)[/tex] is a difference of squares and can be factored as [tex]\( (x - 1)(x + 1) \)[/tex].
So, the fully factored form of the polynomial is:
[tex]\[
x^2(x - 1)(x + 1)(x^2 - 6)
\][/tex]
And there we have it! The polynomial [tex]\( x^6 - 7x^4 + 6x^2 \)[/tex] is factored as [tex]\( x^2(x - 1)(x + 1)(x^2 - 6) \)[/tex].
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