4. Find the GCF of the polynomial, then factor.
8v⁶ + 2v⁵ - 10v⁹
5. Simplify the product of the binomial
[5t + 4]²
6. A rectangle has dimensions 3x 1 and 2x + 5. Write
an expression for the area of the rectangle as a
product and in standard form.
7. Factor m² + 9m - 22
4. To find the greatest common factor (GCF) of the polynomial 8v⁶ + 2v⁵ - 10v⁹, we can first factor out the common factors in each term. The constants 8, 2, and -10 have a common factor of 2, and the variables v⁶, v⁵, and v⁹ have a common factor of v⁵. Therefore, we can factor out 2v⁵ from each term, which gives us:
2v⁵ (4v + 1 - 5v⁴)
Now, let's look at the remaining expression inside the parentheses. We can see that it is a polynomial in terms of v, and we can't factor it further since the exponents of v are not the same. Therefore, the GCF of the polynomial 8v⁶ + 2v⁵ - 10v⁹ is 2v⁵, and the factored form of the polynomial is 2v⁵(4v + 1 - 5v⁴).
5. To simplify the product of the binomial [5t + 4]², we can use the formula for the square of a binomial, which is (a + b)² = a² + 2ab + b².
Plugging in the values from the given binomial [5t + 4], we get:
[5t + 4]² = (5t)² + 2(5t)(4) + 4²
Simplifying further:
[5t + 4]² = 25t² + 40t + 16
So the simplified form of the product of the binomial [5t + 4]² is 25t² + 40t + 16.
6. To write an expression for the area of a rectangle with dimensions 3x - 1 and 2x + 5, we multiply the length and width of the rectangle.
Area = (3x - 1) * (2x + 5)
Using the distributive property, we distribute each term in the first bracket to each term in the second bracket:
Area = 3x * 2x + 3x * 5 - 1 * 2x - 1 * 5
Simplifying further:
Area = 6x² + 15x - 2x - 5
Combining like terms:
Area = 6x² + (15x - 2x) - 5
Simplifying the coefficient of x:
Area = 6x² + 13x - 5
Therefore, the area of the rectangle is represented by the expression 6x² + 13x - 5 in standard form.
7. To factor the quadratic expression m² + 9m - 22, we are looking for two numbers that multiply to -22 and add up to 9 (since the coefficient of m² is 1).
By factoring, we know that the product of the two numbers would be -22, and their sum would be 9.
After trying different pairs of factors, we find that (11, -2) satisfies these conditions.
Therefore, we can rewrite the quadratic expression as:
m² + 11m - 2m - 22
Grouping the terms:
(m² + 11m) - (2m + 22)
Factoring out the common factor from each group:
m(m + 11) - 2(m + 11)
Notice that we now have a common binomial factor of (m + 11). We can factor it out:
(m - 2)(m + 11)
So, the factored form of the quadratic expression m² + 9m - 22 is (m - 2)(m + 11).