Factoring Polynomials and Grouping

The video tutorial explains how to factor quadratic expressions when the leading coefficient is not one, using a method called factoring by grouping. It begins by identifying the coefficients a, b, and c, and then applies the diamond method to find two numbers that multiply to ac and add to b. The middle term is split using these numbers, and the expression is rewritten to allow grouping. Each group is factored separately, and the common binomial factor is identified, leading to the final factored form of the expression.

What is the first step in factoring by grouping when a is not equal to one?

• Solve for x using substitution.

• Multiply the coefficients.

• Directly apply the quadratic formula.

• Identify the coefficients a, b, and c.

What are the values of a, b, and c in the given example?

• a = -21, b = 25, c = 4

• a = 25, b = 4, c = -21

• a = 4, b = 25, c = -21

• a = 1, b = 4, c = 25

In the diamond method, what do you need to find?

• Two numbers that multiply to ac and add to b.

• Two numbers that multiply to c and add to a.

• Two numbers that multiply to b and add to ac.

• Two numbers that multiply to a and add to c.

Which two numbers multiply to -84 and add to 25?

• 28 and -3

• 21 and 4

• 25 and -84

• 3 and -28

What is the purpose of splitting the middle term in factoring by grouping?

• To simplify the equation.

• To find the roots of the equation.

• To create two binomials for grouping.

• To eliminate the constant term.

After splitting the middle term, what is the new polynomial?

• 4x^2 + 28x - 3x - 21

• 4x^2 + 25x - 21

• 4x^2 - 3x + 28x - 21

• 4x^2 + 21x - 28x

When factoring each group, what should you ensure about the parentheses?

• They should contain only variables.

• They should be the same.

• They should contain only constants.

• They should be different.

What do you take out from the first group 4x^2 + 28x?

• 4x

• 7

• x

• 4

What is the final step in the factoring by grouping process?

• Multiply the factors by a constant.

• Verify the solution by expanding the factors.

• Add the factors together.

• Divide the factors by a variable.

What is the final factorization of the polynomial?

• (4x - 3)(x + 7)

• (x + 7)(4x - 3)

• (4x + 3)(x - 7)

• (x - 7)(4x + 3)