1. GCF for 4x^8y^2 + 8x^3y^3:
To find the greatest common factor (GCF), we can factor out the common terms from both terms:
4x^8y^2 + 8x^3y^3 = 4x^3y^2(x^5 + 2y)
2. Factoring the polynomial 7x^3 − 28x^2 + 3x − 12:
To factor by grouping, we group the terms in pairs:
(7x^3 − 28x^2) + (3x − 12)
Taking out the common factor from each pair:
7x^2(x − 4) + 3(x − 4)
Now, we can factor out the common binomial factor:
(x − 4)(7x^2 + 3)
3. Factoring completely: 8x^2 − 14x + 5:
We need to find two numbers that multiply to 5 (the last term) and add up to -14 (the coefficient of the middle term). The numbers that satisfy this condition are -1 and -5. Therefore, we can factor the trinomial as:
8x^2 − 14x + 5 = (2x - 1)(4x - 5)
4. Formula for solving perfect square trinomials:
The formula for a perfect square trinomial is:
(a + b)^2 = a^2 + 2ab + b^2
This formula allows us to expand a perfect square trinomial into its binomial factors.
5. Factored form of x^2 + 14x + 49:
The factored form of x^2 + 14x + 49 is (x + 7)(x + 7), or (x + 7)^2. This is because the trinomial is a perfect square trinomial, where both terms are equal and the square of a binomial.
6. Factoring a^2 – 9:
The expression a^2 – 9 can be factored using the difference of squares formula:
a^2 – 9 = (a + 3)(a - 3)
7. Sketching the graph of the function F(x) = x^3 + x^2 − 9x − 9:
To sketch the graph, we can start by finding the zeros of the function by setting F(x) equal to zero and solving for x. The zeros are the x-values where the graph intersects the x-axis. Next, we can determine the y-intercept by evaluating F(x) when x = 0. Lastly, we can analyze the end behavior by looking at the leading term of the polynomial.
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