So I only have 3 homework questions left and I can't seem to figure out how to solve. I've tried solving them, but I keep getting them wrong. Any help?
Factor the polynomials:
a) 3x^2 +19x - 40
b) x^4 - 25x^2
Simplify the expression completely:
(1/(x-h)^3 - 1/x^3)/8h
b. Difference of two squares, then one of those factors can be factored as a difference of two squares again. I can check your work.
a. What is ( x+8)(3x-5)
c. do some combination.
((x^3-(x-h)^3)/(8h(x^3*(x-h)^3)
now factor the numerator as a difference of two cubes. Doing it in my head, I see some reduction.
ooh, nevermind I figured out how to do a) answer is 3(x-5)(x+8)....I forgot to move the 3 over
and b) uses the difference of squares formula so the answer is x^2(x-5)(x+5)
still can't figure out the third one though!
Of course, I'd be happy to help you with your remaining homework questions! Let's break down the process for solving each question step by step:
a) To factor the quadratic polynomial 3x^2 + 19x - 40, we can use the factoring method.
1. First, check if the polynomial can be factored by looking for common factors among the coefficients. In this case, there are no common factors other than 1.
2. Next, we need to find two numbers that can multiply together to get the product of the leading coefficient (3) multiplied by the constant term (-40), which is -120. These numbers should also add up to the coefficient of the middle term (19).
In this case, we can use the method of trial and error or the quadratic formula to find these numbers. The numbers that work in this case are 4 and -10, since 4*(-10) equals -40 and 4+(-10) equals 19.
3. Now, we rewrite the middle term 19x as the sum of the numbers from the previous step: 4x - 10x.
This gives us the polynomial: 3x^2 + 4x - 10x - 40.
4. Group the terms together by common factors: (3x^2 + 4x) + (-10x - 40).
5. Factor out the greatest common factor from each group: x(3x+4) - 10(3x+4).
6. Notice that we have a common factor of (3x+4) in both groups. We can factor it out: (x-10)(3x+4).
Therefore, the factored form of 3x^2 + 19x - 40 is (x-10)(3x+4).
b) To factor the polynomial x^4 - 25x^2, we can use the factoring method as well.
1. Notice that both terms are perfect squares. So, we can rewrite the expression as (x^2)^2 - (5x)^2.
2. Now, we have a difference of squares, which can be factored using the formula (a^2 - b^2) = (a+b)(a-b).
Applying this formula, we get: (x^2 + 5x)(x^2 - 5x).
3. Now, we can factor out common terms from each grouping: x(x + 5)(x - 5).
Therefore, the factored form of x^4 - 25x^2 is x(x + 5)(x - 5).
c) To simplify the expression (1/(x-h)^3 - 1/x^3)/8h, we need to combine the fractions into a single fraction.
1. Find a common denominator for the two fractions. The least common denominator (LCD) would be (x-h)^3 * x^3.
2. Multiply the numerator and denominator of each fraction by the missing factor from the other one.
This gives us: [x^3 - (x-h)^3]/[(x-h)^3 * x^3 * 8h].
3. Expand the numerator by applying the binomial formula (a-b)^3.
The expansion of (x-h)^3 is x^3 - 3x^2h + 3xh^2 - h^3.
4. Replace the numerator with the expanded form: [x^3 - (x^3 - 3x^2h + 3xh^2 - h^3)]/[(x-h)^3 * x^3 * 8h].
5. Simplify and combine like terms in the numerator: (3x^2h - 3xh^2 + h^3)/[(x-h)^3 * x^3 * 8h].
Therefore, the expression (1/(x-h)^3 - 1/x^3)/8h simplifies to (3x^2h - 3xh^2 + h^3)/[(x-h)^3 * x^3 * 8h].