Is this correct or no. I need help in a couple of problems can someone explain them to me.
Problem#1
directions: Factor each expression
a^2(b-c)-16b^2(b-c)
My answer is:
(b-c)(a-4b)(a+4b)
PROBLEM#2
dIRECTIONS: Find a value for k so that 2x^3-kxy^2 will have the facotors 2x,x-3y,and x+3y
MY steps:
=2x(x-3y)(x+3y)
=2x(x^2-9y^2)
=2x^3-18xy^2
so, k = 18
Problem#3
Directions: Factor each polynomial by grouping the first two terms and the last two terms.
x^3-4x^2+2x-8
MY answer:
(x-4)(x^2+2)
Looks OK!
Problem #1: To factor the expression a^2(b-c)-16b^2(b-c), you correctly recognized that (b-c) is common to both terms. To factor it out, you can write the expression as (b-c) multiplied by the remaining terms: (b-c)(a^2-16b^2).
Next, observe that a^2-16b^2 is a difference of squares, which can be factored as (a+4b)(a-4b).
Putting it all together, you have (b-c)(a+4b)(a-4b) as the fully factored expression.
Problem #2: You were given that 2x^3-kxy^2 has factors of 2x, x-3y, and x+3y. Therefore, you correctly expressed the expression as the product of these factors: 2x(x-3y)(x+3y).
To determine the value of k, you expanded the expression and compared it to 2x^3-kxy^2. By expanding 2x(x-3y)(x+3y), you obtained 2x(x^2-9y^2), which simplifies to 2x^3-18xy^2.
Hence, k is equal to 18, as you correctly concluded, so that 2x^3-kxy^2 equals 2x^3-18xy^2.
Problem #3: To factor the polynomial x^3-4x^2+2x-8 by grouping, you correctly grouped the first two terms (x^3-4x^2) and the last two terms (2x-8).
For the first group, you can factor out the greatest common factor, which is x^2: x^2(x-4).
For the second group, you can factor out the greatest common factor, which is 2: 2(x-4).
Now, notice that both groups have a factor of (x-4). So, you can factor it out: (x-4)(x^2+2).
This completes the factoring, and your answer (x-4)(x^2+2) is correct.
Well done on solving all the problems! If you have any more questions or need further explanation, feel free to ask.