Answer :
Sure! Let's factor each expression step-by-step.
A. Factor [tex]$x^4 - 13x^2 + 36$[/tex] completely:
1. Notice that [tex]$x^4 - 13x^2 + 36$[/tex] can be considered a quadratic in terms of [tex]$x^2$[/tex]. Let's set [tex]$y = x^2$[/tex]. Thus, the expression becomes [tex]$y^2 - 13y + 36$[/tex].
2. Factor [tex]$y^2 - 13y + 36$[/tex] using trial and error or by using the quadratic formula. Upon factoring, we get [tex]$(y - 9)(y - 4)$[/tex].
3. Substitute back [tex]$x^2$[/tex] for [tex]$y$[/tex]. So, we have [tex]$(x^2 - 9)(x^2 - 4)$[/tex].
4. Now, factor these expressions further:
- [tex]$x^2 - 9$[/tex] is a difference of squares and factors as [tex]$(x - 3)(x + 3)$[/tex].
- [tex]$x^2 - 4$[/tex] is also a difference of squares and factors as [tex]$(x - 2)(x + 2)$[/tex].
5. Putting it all together, the complete factorization of [tex]$x^4 - 13x^2 + 36$[/tex] is [tex]$(x - 3)(x + 3)(x - 2)(x + 2)$[/tex].
B. Factor [tex]$4x^4 - 13x^2 + 9$[/tex] completely:
1. First, observe that this is in a quadratic form in terms of [tex]$x^2$[/tex]. Let's set [tex]$y = x^2$[/tex]. The expression then becomes [tex]$4y^2 - 13y + 9$[/tex].
2. To factor [tex]$4y^2 - 13y + 9$[/tex], we look for two numbers that multiply to [tex]$4 \times 9 = 36$[/tex] and add up to [tex]$-13$[/tex]. The numbers [tex]$-9$[/tex] and [tex]$-4$[/tex] work for this.
3. Split the middle term using [tex]$-9$[/tex] and [tex]$-4$[/tex]: [tex]$4y^2 - 9y - 4y + 9$[/tex].
4. Group the terms: [tex]$(4y^2 - 9y) + (-4y + 9)$[/tex].
5. Factor by grouping:
- [tex]$4y - 3$[/tex] is a common factor of the first group, giving us [tex]$y(4y - 3)$[/tex].
- [tex]$1(4y - 3)$[/tex] is a factor in the second group.
6. Combine these to get [tex]$(2y - 3)(2y - 3)$[/tex], which is [tex]$(2y - 3)^2$[/tex].
7. Replace [tex]$y$[/tex] back with [tex]$x^2$[/tex] to get [tex]$(2x^2 - 3)^2$[/tex].
8. Factor [tex]$2x^2 - 3$[/tex] into its factors: [tex]$(2x - 3)(2x + 3)$[/tex].
9. So, the complete factorization of [tex]$4x^4 - 13x^2 + 9$[/tex] is [tex]$(x - 1)(x + 1)(2x - 3)(2x + 3)$[/tex].
I hope this explanation helps you understand how to factor these expressions completely!
A. Factor [tex]$x^4 - 13x^2 + 36$[/tex] completely:
1. Notice that [tex]$x^4 - 13x^2 + 36$[/tex] can be considered a quadratic in terms of [tex]$x^2$[/tex]. Let's set [tex]$y = x^2$[/tex]. Thus, the expression becomes [tex]$y^2 - 13y + 36$[/tex].
2. Factor [tex]$y^2 - 13y + 36$[/tex] using trial and error or by using the quadratic formula. Upon factoring, we get [tex]$(y - 9)(y - 4)$[/tex].
3. Substitute back [tex]$x^2$[/tex] for [tex]$y$[/tex]. So, we have [tex]$(x^2 - 9)(x^2 - 4)$[/tex].
4. Now, factor these expressions further:
- [tex]$x^2 - 9$[/tex] is a difference of squares and factors as [tex]$(x - 3)(x + 3)$[/tex].
- [tex]$x^2 - 4$[/tex] is also a difference of squares and factors as [tex]$(x - 2)(x + 2)$[/tex].
5. Putting it all together, the complete factorization of [tex]$x^4 - 13x^2 + 36$[/tex] is [tex]$(x - 3)(x + 3)(x - 2)(x + 2)$[/tex].
B. Factor [tex]$4x^4 - 13x^2 + 9$[/tex] completely:
1. First, observe that this is in a quadratic form in terms of [tex]$x^2$[/tex]. Let's set [tex]$y = x^2$[/tex]. The expression then becomes [tex]$4y^2 - 13y + 9$[/tex].
2. To factor [tex]$4y^2 - 13y + 9$[/tex], we look for two numbers that multiply to [tex]$4 \times 9 = 36$[/tex] and add up to [tex]$-13$[/tex]. The numbers [tex]$-9$[/tex] and [tex]$-4$[/tex] work for this.
3. Split the middle term using [tex]$-9$[/tex] and [tex]$-4$[/tex]: [tex]$4y^2 - 9y - 4y + 9$[/tex].
4. Group the terms: [tex]$(4y^2 - 9y) + (-4y + 9)$[/tex].
5. Factor by grouping:
- [tex]$4y - 3$[/tex] is a common factor of the first group, giving us [tex]$y(4y - 3)$[/tex].
- [tex]$1(4y - 3)$[/tex] is a factor in the second group.
6. Combine these to get [tex]$(2y - 3)(2y - 3)$[/tex], which is [tex]$(2y - 3)^2$[/tex].
7. Replace [tex]$y$[/tex] back with [tex]$x^2$[/tex] to get [tex]$(2x^2 - 3)^2$[/tex].
8. Factor [tex]$2x^2 - 3$[/tex] into its factors: [tex]$(2x - 3)(2x + 3)$[/tex].
9. So, the complete factorization of [tex]$4x^4 - 13x^2 + 9$[/tex] is [tex]$(x - 1)(x + 1)(2x - 3)(2x + 3)$[/tex].
I hope this explanation helps you understand how to factor these expressions completely!
