8.5. Multiplying a Polynomial by a Polynomial

Let us multiply one binomial (2a + 3b) by another binomial, say (3a + 4b). We do this step-by-step, as we did in earlier cases, following the distributive law of multiplication,

(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b) = (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b) = 6a² + 9ab + 8ba + 12b² = 6a² + 17ab + 12b² (Since ba = ab).

When we carry out term by term multiplication, we expect 2 × 2 = 4 terms to be present. But two of these are like terms, which are combined, and hence we get 3 terms.

To multiply two binomials, we apply the distributive property. This means that every term in the first binomial multiplies every term in the second binomial. In our example, we multiplied each term in (3a + 4b) by each term in (2a + 3b). This first gives us four products, and then we combine any like terms to simplify the expression. The key point here is to remember to distribute each term properly.

Imagine you have two bags of fruits: one bag contains 2 apples and 3 oranges, while the other bag contains 3 apples and 4 oranges. If you combine the fruits from both bags, you would not just count the apples and oranges separately; instead, you would see how many apples you have in total and how many oranges, just like combining terms in our multiplication.

Example 8: Multiply

(i) (x – 4) and (2x + 3) Solution: (i) (x – 4) × (2x + 3) = x × (2x + 3) – 4 × (2x + 3) = (x × 2x) + (x × 3) – (4 × 2x) – (4 × 3) = 2x² + 3x – 8x – 12 = 2x² – 5x – 12 (Adding like terms)

(ii) (x – y) and (3x + 5y) = x × (3x + 5y) – y × (3x + 5y) = (x × 3x) + (x × 5y) – (y × 3x) – (y × 5y) = 3x² + 5xy – 3yx – 5y² = 3x² + 2xy – 5y² (Adding like terms)

The examples demonstrate how to multiply binomials by applying the distributive property step-by-step. For the first example, we take each term of (x – 4) and multiply it by each term of (2x + 3), which results in several products, some of which are like terms. After combining like terms, we arrive at the final polynomial. The same method applies to the second example, reinforcing the understanding of this multiplication process.

Think of it like packing boxes. You have a box labeled 'x - 4' for small items and another box labeled '2x + 3' for larger items. When packing, you want to see how many combinations of small and large items you can fit into a bigger box. Each combination gives you a product, similar to how we multiply terms in algebra.

In this multiplication, we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial. We shall get in all 3 × 2 = 6 terms, which may reduce to 5 or less, if the term by term multiplication results in like terms. Consider (a + 7) × (a² + 3a + 5) = a × (a² + 3a + 5) + 7 × (a² + 3a + 5) = a³ + 3a² + 5a + 7a² + 21a + 35 = a³ + (3a² + 7a²) + (5a + 21a) + 35 = a³ + 10a² + 26a + 35.

When multiplying a binomial and a trinomial, you multiply each term in the binomial by each term in the trinomial, resulting in multiple terms. In our example, we work through the multiplication systematically, combining like terms afterward. This emphasizes the importance of organizing the operation and being attentive to like terms.

Visualize mixing paint. You have two colors represented by the binomial and three shades by the trinomial. When you mix these, you get a range of new shades (terms) from each combination. Some shades may look similar (like terms) and can be blended together, which represents simplifying the polynomial after multiplication.

Example 10: Simplify (a + b) (2a – 3b + c) – (2a – 3b) c. Solution: We have (a + b) (2a – 3b + c) = a (2a – 3b + c) + b (2a – 3b + c) = 2a² – 3ab + ac + 2ab – 3b² + bc = 2a² – ab – 3b² + bc + ac (Note, –3ab and 2ab are like terms) and (2a – 3b) c = 2ac – 3bc. Therefore, (a + b) (2a – 3b + c) – (2a – 3b) c = 2a² – ab – 3b² + bc + ac – (2ac – 3bc) = 2a² – ab – 3b² + (bc + 3bc) + (ac – 2ac) = 2a² – 3b² – ab + 4bc – ac.

This example illustrates how to simplify expressions involving polynomial products by organizing steps and carefully combining like terms. First, we distribute both terms in the binomial to each term in the trinomial, followed by managing the additional polynomial and resulting expressions. The focus should be on accuracy and clarity in combining like terms before arriving at the final simplified result.

Picture a recipe where you first combine flavors (the binomial) with multiple ingredients (the trinomial). After mixing, you may need to adjust the flavors by adding or subtracting certain ingredients to reach a delicious outcome! Just like cooking, careful combination and adjustments lead to a refined result.

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