Answer :
Sure! Let's factorize the given expressions step by step.
### 3. Factorize [tex]\( 14x^4 - 45x^2 - 14 \)[/tex]
1. Identify the expression: We are given [tex]\( 14x^4 - 45x^2 - 14 \)[/tex].
2. Substitute [tex]\( y = x^2 \)[/tex]: This simplifies the quartic equation into a quadratic form:
[tex]\[
14y^2 - 45y - 14
\][/tex]
3. Find the factor pairs: We look for two numbers that multiply to [tex]\( 14 \times -14 = -196 \)[/tex] and add up to [tex]\(-45\)[/tex].
[tex]\[
Those numbers are -49 and +4, because -49 \times 4 = -196 \quad \text{and} \quad -49 + 4 = -45.
\][/tex]
4. Rewrite the quadratic: Using these factor pairs, we rewrite the quadratic:
[tex]\[
14y^2 - 49y + 4y - 14
\][/tex]
5. Factor by grouping:
[tex]\[
(14y^2 - 49y) + (4y - 14)
\][/tex]
[tex]\[
7y(2y - 7) + 2(2y - 7)
\][/tex]
[tex]\[
(7y + 2)(2y - 7)
\][/tex]
6. Substitute [tex]\( y = x^2 \)[/tex] back:
[tex]\[
(7x^2 + 2)(2x^2 - 7)
\][/tex]
So the factorized form of [tex]\( 14x^4 - 45x^2 - 14 \)[/tex] is:
[tex]\[
(2x^2 - 7)(7x^2 + 2)
\][/tex]
### 4. Factorize [tex]\( 243 - 32b^5 \)[/tex]
1. Identify the expression: We are given [tex]\( 243 - 32b^5 \)[/tex].
2. Rewrite the expression: Notice that 243 is a power of 3, and 32 is a power of 2:
[tex]\[
243 = 3^5 \quad \text{and} \quad 32b^5 = (2b)^5
\][/tex]
3. Use the difference of powers formula:
[tex]\[
a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)
\][/tex]
Let [tex]\( a = 3 \)[/tex] and [tex]\( b = 2b \)[/tex]:
[tex]\[
(3)^5 - (2b)^5 = (3 - 2b)(3^4 + 3^3(2b) + 3^2(2b)^2 + 3(2b)^3 + (2b)^4)
\][/tex]
4. Simplify the terms:
[tex]\[
= (3 - 2b)(81 + 3^3 \cdot 2b + 3^2 \cdot 4b^2 + 3 \cdot 8b^3 + 16b^4)
\][/tex]
[tex]\[
= (3 - 2b)(81 + 54b + 36b^2 + 24b^3 + 16b^4)
\][/tex]
So the factorized form of [tex]\( 243 - 32b^5 \)[/tex] is:
[tex]\[
-(2b - 3)(16b^4 + 24b^3 + 36b^2 + 54b + 81)
\][/tex]
Note the negative sign to match the original expression's coefficients correctly.
### Summary
The factorizations are:
1. [tex]\( 14x^4 - 45x^2 - 14 \)[/tex]: [tex]\( (2x^2 - 7)(7x^2 + 2) \)[/tex]
2. [tex]\( 243 - 32b^5 \)[/tex]: [tex]\( -(2b - 3)(16b^4 + 24b^3 + 36b^2 + 54b + 81) \)[/tex]
I hope this helps! If you have any additional questions, feel free to ask.
### 3. Factorize [tex]\( 14x^4 - 45x^2 - 14 \)[/tex]
1. Identify the expression: We are given [tex]\( 14x^4 - 45x^2 - 14 \)[/tex].
2. Substitute [tex]\( y = x^2 \)[/tex]: This simplifies the quartic equation into a quadratic form:
[tex]\[
14y^2 - 45y - 14
\][/tex]
3. Find the factor pairs: We look for two numbers that multiply to [tex]\( 14 \times -14 = -196 \)[/tex] and add up to [tex]\(-45\)[/tex].
[tex]\[
Those numbers are -49 and +4, because -49 \times 4 = -196 \quad \text{and} \quad -49 + 4 = -45.
\][/tex]
4. Rewrite the quadratic: Using these factor pairs, we rewrite the quadratic:
[tex]\[
14y^2 - 49y + 4y - 14
\][/tex]
5. Factor by grouping:
[tex]\[
(14y^2 - 49y) + (4y - 14)
\][/tex]
[tex]\[
7y(2y - 7) + 2(2y - 7)
\][/tex]
[tex]\[
(7y + 2)(2y - 7)
\][/tex]
6. Substitute [tex]\( y = x^2 \)[/tex] back:
[tex]\[
(7x^2 + 2)(2x^2 - 7)
\][/tex]
So the factorized form of [tex]\( 14x^4 - 45x^2 - 14 \)[/tex] is:
[tex]\[
(2x^2 - 7)(7x^2 + 2)
\][/tex]
### 4. Factorize [tex]\( 243 - 32b^5 \)[/tex]
1. Identify the expression: We are given [tex]\( 243 - 32b^5 \)[/tex].
2. Rewrite the expression: Notice that 243 is a power of 3, and 32 is a power of 2:
[tex]\[
243 = 3^5 \quad \text{and} \quad 32b^5 = (2b)^5
\][/tex]
3. Use the difference of powers formula:
[tex]\[
a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)
\][/tex]
Let [tex]\( a = 3 \)[/tex] and [tex]\( b = 2b \)[/tex]:
[tex]\[
(3)^5 - (2b)^5 = (3 - 2b)(3^4 + 3^3(2b) + 3^2(2b)^2 + 3(2b)^3 + (2b)^4)
\][/tex]
4. Simplify the terms:
[tex]\[
= (3 - 2b)(81 + 3^3 \cdot 2b + 3^2 \cdot 4b^2 + 3 \cdot 8b^3 + 16b^4)
\][/tex]
[tex]\[
= (3 - 2b)(81 + 54b + 36b^2 + 24b^3 + 16b^4)
\][/tex]
So the factorized form of [tex]\( 243 - 32b^5 \)[/tex] is:
[tex]\[
-(2b - 3)(16b^4 + 24b^3 + 36b^2 + 54b + 81)
\][/tex]
Note the negative sign to match the original expression's coefficients correctly.
### Summary
The factorizations are:
1. [tex]\( 14x^4 - 45x^2 - 14 \)[/tex]: [tex]\( (2x^2 - 7)(7x^2 + 2) \)[/tex]
2. [tex]\( 243 - 32b^5 \)[/tex]: [tex]\( -(2b - 3)(16b^4 + 24b^3 + 36b^2 + 54b + 81) \)[/tex]
I hope this helps! If you have any additional questions, feel free to ask.
