Algebra I : Semester II : Polynomials

Sections:

Introduction  |   Section 1  |   Section 2  |   Section 3  |  Section 4  |  Section 5  |   Section 6

Section Three

Part 1  |  Part 2   |  Part 3  |  Part 4  |  Part 5

Algebra 1: Section 3: Polynomials Multiplying Polynomials by a Monomial You used multiplication of a binomial by a constant when you learned the distributive property. For instance: Example 1:  Finding Area Find the area of the rectangle. Use the area formula for a rectangle. A = LW                 Formula A = 3(2n + 8)        Substitute A = 3(2n) + 3(8)    Use the distributive property A = 6n + 24          Simplify Now what if the rectangle had the measures pictured in the presentation below for the side? Take a look at this in the presentation.   Click here for a presentation on Multiplying Polynomials by Monomials. Now let’s sum up what we just learned.   To Multiply a Polynomial by a Monomial Distribute the monomial times each term in the polynomial. Multiply the coefficients and add exponents of common bases. Example 2: Multiply Polynomials Simplify -3x2(4x2 – x + 5). = (-3x2)(4x2) + (-3x2)(-x) + (-3x2)(5)    Distribute = -12x4 + 3x3 + -15x2                       Multiply coefficients and add exponents = -12x4 + 3x3 – 15x2                        Simplify Now that you know how to multiply by distributing, look at expressions in which you must distribute before you can simplify the expressions. Example 3: Simplify Expressions Simplify 2x(3x – 5) + 4x(x + 2). = 2x(3x) + 2x(-5) + 4x(x) + 4x(2)     Distribute = 6x2 – 10x + 4x2 + 8x                     Multiply coefficients and add exponents = (6x2 + 4x2) + (-10x + 8x)               Group into like terms = 10x2 – 2x                                     Add like terms Example 4: Solve Equations Solve:  6a(a – 3) = 3a(2a – 12) – 36 6a(a – 3) = 3a(2a – 12) – 36   6a2 – 18a = 6a2 –  36a – 36 Distribute 6a2 – 18a – 6a2 = 6a2 –  36a – 36 – 6a2 Subtract 6a2 from both sides -18a = -36a – 36 Simplify -18a + 36a = -36a – 36 + 36a Add 36a to both sides 18a = -36 Simplify a = -36/18 Divide by 18 a = -2 Simplify Therefore a = -2. You can check by substituting -2 for each a in the original equation. Check: 6(-2)(-2 – 3) = 3(-2)[2(-2) – 12] – 36 -12(-5) = -6(-16) – 36 60 = 96 – 36 60 = 60  √ Example 5: Area Problems Find the area of the given figure. There is not an area formula for this kind of shape. However, you can divide the shape into two rectangular regions. This way, you can find the area of each and add them together. Area A = x(x + 2) = x2 + 2x Area B = 2x(2x + 2) = 4x2 + 4x Total Region = (x2 + 2x) + (4x2 + 4x) = 5x2 + 6x   Practice Multiply. 3(2x2 + 5y – 4) = 6x2 + 15y – 12 –(2x3 – x2 + 5x) = -2x3 + x2 – 5x 4x3(x2 + 3x – 2) = 4x5 + 12x4 – 8x3 3x2(x2 – 2xy+ y2) = 3x4 – 6x3y + 3x2y2  ¼ a(4a2 + 8a – 12) = a3 + 2a2 – 3a -2m2n(3m3 – 5mn + 4n3) = -6m5n + 10m3n2 – 8m2n4 Simplify. 5n(2n + 4) – 3n(n – 2) = 10n2 + 20n – 3n2 + 6n = 7n2 + 26n -2a(a2 – 5) + 3a(a2 – 6) = -2a3 + 10a + 3a3 – 18a = a3 – 8a 2x(6x – 9) – (8x + 6) = 12x2 – 18x – 8x – 6 = 12x2 – 26x – 6  2a(a2 – a + 2) + a(a3 – 5a2 – 7a) = 2a3 – 2a2 + 4a + a4 – 5a3 – 7a2 = a4 – 3a3 – 9a2 + 4a Solve.  5(x – 2) – 3x = 10 5x – 10 – 3x = 10 Distribute 2x – 10 = 10 5x – 3x 2x = 20 Add 10 x = 10 Divide by 2  1/3 (6 – 3n) = -12 2 – n = -12 Distribute -n = -14 Subtract 2 n = 14 Divide by -1  5x(x – 2) = 5x2 + 40 5x2 – 10x = 5x2 + 40 Distribute -10x = 40 Subtract 5x2 x = -4 Divide by -10  3(2y + 5) – 4(1 – 5y) = 9(3 + 2y) 6y + 15 – 4 + 20y = 27 + 18y Distribute 26y + 11 = 27 + 18y Simplify by adding like terms 8y + 11 = 27 Subtract 18y 8y = 16 Subtract 11 y = 2 Divide by 8  2(5n – 4) – 3(n – 5) = 8(2n – 7) 10n – 8 – 3n + 15 = 16n – 56 Distribute 7n + 7 = 16n - 56 Simplify by adding like terms -9n + 7 = -56 Subtract 16n -9n = -63 Subtract 7 n = 7 Divide by -9   Find the area of the shaded region. Divide the region into two rectangular regions. Then find each area and add. Area A = x(3x – 7) = 3x2 – 7x Area B = 2x(5x + 2) = 10x2 + 4x Total Area = (3x2 – 7x) + (10x2 + 4x) = 13x2 – 3x   Find the area of the shaded region. There are several ways to do this, but the simplest is to name the empty rectangular space A and the larger rectangle B. Then find the area and subtract B – A. Area A = 5(3n + 2) = 15n + 10 Area B = 8(5n – 1) = 40n – 8 Area B – A = (40n – 8) – (15n + 10)                  = 40n – 8 – 15n – 10                  = 25n – 18 Now go on to the next part

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