Answer :
To factor the expression [tex]\( 25y^6 - 40y^5 + 16y^4 \)[/tex] completely, let's follow these steps:
1. Identify the Greatest Common Factor (GCF):
Look for the greatest common factor in all the terms of the polynomial.
- The coefficients are 25, 40, and 16. The GCF of these numbers is 1.
- The smallest power of [tex]\( y \)[/tex] is [tex]\( y^4 \)[/tex], so [tex]\( y^4 \)[/tex] will be part of the GCF.
Thus, the GCF of the expression is [tex]\( y^4 \)[/tex].
2. Factor out the GCF:
Divide each term in the polynomial by the GCF [tex]\( y^4 \)[/tex]:
[tex]\[
25y^6 - 40y^5 + 16y^4 = y^4(25y^2 - 40y + 16)
\][/tex]
3. Factor the quadratic [tex]\( 25y^2 - 40y + 16 \)[/tex]:
Now, we need to factor the quadratic expression inside the parentheses: [tex]\( 25y^2 - 40y + 16 \)[/tex].
To factor the quadratic, look for two numbers that multiply to [tex]\( 25 \times 16 = 400 \)[/tex] and add up to [tex]\(-40\)[/tex].
The numbers [tex]\(-20\)[/tex] and [tex]\(-20\)[/tex] work, as [tex]\((-20) \times (-20) = 400\)[/tex] and [tex]\((-20) + (-20) = -40\)[/tex].
This allows us to rewrite the quadratic as:
[tex]\[
25y^2 - 20y - 20y + 16 = 5y(5y - 4) - 4(5y - 4)
\][/tex]
Factoring by grouping:
[tex]\[
(5y - 4)(5y - 4) = (5y - 4)^2
\][/tex]
4. Combine the factors:
Substitute this back into the expression:
[tex]\[
y^4(25y^2 - 40y + 16) = y^4(5y - 4)^2
\][/tex]
Hence, the completely factored form of the polynomial [tex]\( 25y^6 - 40y^5 + 16y^4 \)[/tex] is:
[tex]\[
y^4(5y - 4)^2
\][/tex]
1. Identify the Greatest Common Factor (GCF):
Look for the greatest common factor in all the terms of the polynomial.
- The coefficients are 25, 40, and 16. The GCF of these numbers is 1.
- The smallest power of [tex]\( y \)[/tex] is [tex]\( y^4 \)[/tex], so [tex]\( y^4 \)[/tex] will be part of the GCF.
Thus, the GCF of the expression is [tex]\( y^4 \)[/tex].
2. Factor out the GCF:
Divide each term in the polynomial by the GCF [tex]\( y^4 \)[/tex]:
[tex]\[
25y^6 - 40y^5 + 16y^4 = y^4(25y^2 - 40y + 16)
\][/tex]
3. Factor the quadratic [tex]\( 25y^2 - 40y + 16 \)[/tex]:
Now, we need to factor the quadratic expression inside the parentheses: [tex]\( 25y^2 - 40y + 16 \)[/tex].
To factor the quadratic, look for two numbers that multiply to [tex]\( 25 \times 16 = 400 \)[/tex] and add up to [tex]\(-40\)[/tex].
The numbers [tex]\(-20\)[/tex] and [tex]\(-20\)[/tex] work, as [tex]\((-20) \times (-20) = 400\)[/tex] and [tex]\((-20) + (-20) = -40\)[/tex].
This allows us to rewrite the quadratic as:
[tex]\[
25y^2 - 20y - 20y + 16 = 5y(5y - 4) - 4(5y - 4)
\][/tex]
Factoring by grouping:
[tex]\[
(5y - 4)(5y - 4) = (5y - 4)^2
\][/tex]
4. Combine the factors:
Substitute this back into the expression:
[tex]\[
y^4(25y^2 - 40y + 16) = y^4(5y - 4)^2
\][/tex]
Hence, the completely factored form of the polynomial [tex]\( 25y^6 - 40y^5 + 16y^4 \)[/tex] is:
[tex]\[
y^4(5y - 4)^2
\][/tex]
