How can factoring by grouping be used to express a polynomial as a product of two binomials?

You're on a roll — 2.5 To express a polynomial as a product of two binomials after factoring by grouping, you use the distributive property. If you have an expression like x^2(x + 2) + 3(x + 2), the common factor is (x + 2). You factor this common binomial out: x^2(x + 2) + 3(x + 2) = (x + 2)(x^2 + 3) This results in a product of two binomials. 2.6 Joseph wants to factor an algebraic expression (not visible). Here is a three-step guide for factoring by grouping, which is often used for four-term polynomials: Step 1: Group the terms. Divide the four-term polynomial into two pairs of terms. For example, (ax^3 + bx^2) + (cx + d). Step 2: Factor out the greatest common factor (GCF) from each group. This should result in a common binomial factor in both groups. For example, x^2(ax + b) + c(ax + b). Step 3: Factor out the common binomial factor. Use the distributive property to factor out the common binomial, expressing the polynomial as a product of two binomials. For example, (ax + b)(x^2 + c). ACTIVITY 3 3.1.1 The table in 3.1.2 indicates that we are looking for two integers whose product is 6 and whose sum is 5. For a trinomial in the form x^2 + bx + c, the product of the two integers is c and their sum is b. Therefore, the values of b and c in the trinomial are: b = 5 c = 6 3.1.2 Use the table below to determine the two integers. We need two integers whose product is 6 and whose sum is 5. | Factors of 6 | Product of the two factors | Sum of the two factors | Result | | :----------- | :------------------------- | :--------------------- | :----- | | 1 and 6 | 1 × 6 = 6 | 1 + 6 = 7 | No | | 2 and 3 | 2 × 3 = 6 | 2 + 3 = 5 | Yes | | -1 and -6 | (-1) × (-6) = 6 | -1 + (-6) = -7 | No | | -2 and -3 | (-2) × (-3) = 6 | -2 + (-3) = -5 | No | The two integers are 2 and 3. Drop the next question.