Answer :
To factor the expression [tex]\(2x^4 - 21x^2 - 11\)[/tex], we'll go through a step-by-step process to factor it completely.
1. Identify the Structure: Notice that the given expression can be seen as a quadratic in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex], so the expression becomes:
[tex]\[
2y^2 - 21y - 11
\][/tex]
2. Factor the Quadratic Polynomial: We are looking to factor this into two binomials. We need to find two numbers that multiply to [tex]\(2 \times -11 = -22\)[/tex] and add to [tex]\(-21\)[/tex].
3. Find the Numbers: After examining the factors of [tex]\(-22\)[/tex] (which are 1 & -22, and 2 & -11), we see that:
[tex]\[
2y^2 - 21y - 11 = (2y + 1)(y - 11)
\][/tex]
This factorization works because [tex]\( (2 \times -11) + (1 \times 1) = -22 + 1 = -21\)[/tex].
4. Substitute Back: Replace [tex]\(y\)[/tex] with [tex]\(x^2\)[/tex] in the factored form:
[tex]\[
2(x^2) + 1\quad \text{and} \quad (x^2 - 11)
\][/tex]
Which gives us:
[tex]\[
(2x^2 + 1)(x^2 - 11)
\][/tex]
5. Verify: It's always a good idea to verify your factorization by expanding the factors to see if we get back the original expression.
- Multiply the two factors:
[tex]\[
(2x^2 + 1)(x^2 - 11) = 2x^4 - 22x^2 + x^2 - 11 = 2x^4 - 21x^2 - 11
\][/tex]
- The original expression is restored, confirming our factorization is correct.
The complete factorization of [tex]\(2x^4 - 21x^2 - 11\)[/tex] is:
[tex]\[
(2x^2 + 1)(x^2 - 11)
\][/tex]
1. Identify the Structure: Notice that the given expression can be seen as a quadratic in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex], so the expression becomes:
[tex]\[
2y^2 - 21y - 11
\][/tex]
2. Factor the Quadratic Polynomial: We are looking to factor this into two binomials. We need to find two numbers that multiply to [tex]\(2 \times -11 = -22\)[/tex] and add to [tex]\(-21\)[/tex].
3. Find the Numbers: After examining the factors of [tex]\(-22\)[/tex] (which are 1 & -22, and 2 & -11), we see that:
[tex]\[
2y^2 - 21y - 11 = (2y + 1)(y - 11)
\][/tex]
This factorization works because [tex]\( (2 \times -11) + (1 \times 1) = -22 + 1 = -21\)[/tex].
4. Substitute Back: Replace [tex]\(y\)[/tex] with [tex]\(x^2\)[/tex] in the factored form:
[tex]\[
2(x^2) + 1\quad \text{and} \quad (x^2 - 11)
\][/tex]
Which gives us:
[tex]\[
(2x^2 + 1)(x^2 - 11)
\][/tex]
5. Verify: It's always a good idea to verify your factorization by expanding the factors to see if we get back the original expression.
- Multiply the two factors:
[tex]\[
(2x^2 + 1)(x^2 - 11) = 2x^4 - 22x^2 + x^2 - 11 = 2x^4 - 21x^2 - 11
\][/tex]
- The original expression is restored, confirming our factorization is correct.
The complete factorization of [tex]\(2x^4 - 21x^2 - 11\)[/tex] is:
[tex]\[
(2x^2 + 1)(x^2 - 11)
\][/tex]
