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Answer :
To factor the expression [tex]\(x^4 + 4x^2 - 45\)[/tex] completely, let's break the expression down step by step.
1. Identify the structure: We notice that [tex]\(x^4 + 4x^2 - 45\)[/tex] is a polynomial in terms of [tex]\(x^2\)[/tex]. This suggests that we can use substitution to make it easier to handle.
2. Substitution: Let [tex]\( y = x^2 \)[/tex]. This changes our expression from [tex]\( x^4 + 4x^2 - 45 \)[/tex] to [tex]\( y^2 + 4y - 45 \)[/tex].
3. Factor the quadratic in terms of [tex]\(y\)[/tex]: We now look to factor [tex]\( y^2 + 4y - 45 \)[/tex]. We need two numbers that multiply to [tex]\(-45\)[/tex] and add to [tex]\(4\)[/tex]. These numbers are [tex]\(9\)[/tex] and [tex]\(-5\)[/tex].
4. Write the factored form: Using these numbers, we can write the expression as:
[tex]\[
y^2 + 4y - 45 = (y + 9)(y - 5)
\][/tex]
5. Back-substitute [tex]\( y = x^2 \)[/tex]: Replace [tex]\( y \)[/tex] with [tex]\( x^2 \)[/tex] to return to the original variable:
[tex]\[
(x^2 + 9)(x^2 - 5)
\][/tex]
6. Check for further factoring: Both [tex]\( x^2 + 9 \)[/tex] and [tex]\( x^2 - 5 \)[/tex] cannot be factored further using real numbers (since [tex]\( x^2 + 9 \)[/tex] involves a sum of squares, which doesn't factor neatly with real numbers without involving imaginary numbers).
So, the complete factorization of [tex]\(x^4 + 4x^2 - 45\)[/tex] over the real numbers is:
[tex]\[
(x^2 + 9)(x^2 - 5)
\][/tex]
That's the expression factored completely!
1. Identify the structure: We notice that [tex]\(x^4 + 4x^2 - 45\)[/tex] is a polynomial in terms of [tex]\(x^2\)[/tex]. This suggests that we can use substitution to make it easier to handle.
2. Substitution: Let [tex]\( y = x^2 \)[/tex]. This changes our expression from [tex]\( x^4 + 4x^2 - 45 \)[/tex] to [tex]\( y^2 + 4y - 45 \)[/tex].
3. Factor the quadratic in terms of [tex]\(y\)[/tex]: We now look to factor [tex]\( y^2 + 4y - 45 \)[/tex]. We need two numbers that multiply to [tex]\(-45\)[/tex] and add to [tex]\(4\)[/tex]. These numbers are [tex]\(9\)[/tex] and [tex]\(-5\)[/tex].
4. Write the factored form: Using these numbers, we can write the expression as:
[tex]\[
y^2 + 4y - 45 = (y + 9)(y - 5)
\][/tex]
5. Back-substitute [tex]\( y = x^2 \)[/tex]: Replace [tex]\( y \)[/tex] with [tex]\( x^2 \)[/tex] to return to the original variable:
[tex]\[
(x^2 + 9)(x^2 - 5)
\][/tex]
6. Check for further factoring: Both [tex]\( x^2 + 9 \)[/tex] and [tex]\( x^2 - 5 \)[/tex] cannot be factored further using real numbers (since [tex]\( x^2 + 9 \)[/tex] involves a sum of squares, which doesn't factor neatly with real numbers without involving imaginary numbers).
So, the complete factorization of [tex]\(x^4 + 4x^2 - 45\)[/tex] over the real numbers is:
[tex]\[
(x^2 + 9)(x^2 - 5)
\][/tex]
That's the expression factored completely!
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