Answer :
Sure! Let's factor the expression [tex]\(x^4 + 4x^2 - 45\)[/tex] completely.
1. Identify the expression structure: The expression is a polynomial in terms of [tex]\(x\)[/tex]. It has three terms, and the highest degree of [tex]\(x\)[/tex] is 4.
2. Substitution for simplification: Notice that the middle term is [tex]\(4x^2\)[/tex]. This suggests that we can make a substitution to simplify the structure. Let's set [tex]\(u = x^2\)[/tex]. This transforms the expression into:
[tex]\[
u^2 + 4u - 45
\][/tex]
3. Factor the quadratic expression: Now, we factor the quadratic [tex]\(u^2 + 4u - 45\)[/tex]. Look for two numbers whose product is [tex]\(-45\)[/tex] and sum is [tex]\(4\)[/tex]. These numbers are [tex]\(9\)[/tex] and [tex]\(-5\)[/tex].
4. Write the expression using these numbers: Rewrite the quadratic as:
[tex]\[
u^2 + 9u - 5u - 45
\][/tex]
5. Factor by grouping: Group the terms and factor each group:
[tex]\[
(u^2 + 9u) - (5u + 45)
\][/tex]
Factor out common factors in each group:
[tex]\[
u(u + 9) - 5(u + 9)
\][/tex]
6. Combine the groups: Notice that [tex]\((u + 9)\)[/tex] is a common factor:
[tex]\[
(u - 5)(u + 9)
\][/tex]
7. Substitute back: Remember that [tex]\(u = x^2\)[/tex]. Substitute back to get the final factored form:
[tex]\[
(x^2 - 5)(x^2 + 9)
\][/tex]
This is the completely factored form of the original expression [tex]\(x^4 + 4x^2 - 45\)[/tex].
1. Identify the expression structure: The expression is a polynomial in terms of [tex]\(x\)[/tex]. It has three terms, and the highest degree of [tex]\(x\)[/tex] is 4.
2. Substitution for simplification: Notice that the middle term is [tex]\(4x^2\)[/tex]. This suggests that we can make a substitution to simplify the structure. Let's set [tex]\(u = x^2\)[/tex]. This transforms the expression into:
[tex]\[
u^2 + 4u - 45
\][/tex]
3. Factor the quadratic expression: Now, we factor the quadratic [tex]\(u^2 + 4u - 45\)[/tex]. Look for two numbers whose product is [tex]\(-45\)[/tex] and sum is [tex]\(4\)[/tex]. These numbers are [tex]\(9\)[/tex] and [tex]\(-5\)[/tex].
4. Write the expression using these numbers: Rewrite the quadratic as:
[tex]\[
u^2 + 9u - 5u - 45
\][/tex]
5. Factor by grouping: Group the terms and factor each group:
[tex]\[
(u^2 + 9u) - (5u + 45)
\][/tex]
Factor out common factors in each group:
[tex]\[
u(u + 9) - 5(u + 9)
\][/tex]
6. Combine the groups: Notice that [tex]\((u + 9)\)[/tex] is a common factor:
[tex]\[
(u - 5)(u + 9)
\][/tex]
7. Substitute back: Remember that [tex]\(u = x^2\)[/tex]. Substitute back to get the final factored form:
[tex]\[
(x^2 - 5)(x^2 + 9)
\][/tex]
This is the completely factored form of the original expression [tex]\(x^4 + 4x^2 - 45\)[/tex].
