Polynomials and Rational Expressions

Polynomial Operations

  • Polynomial Addition and Subtraction:

    • Combine like terms by adding or subtracting their respective coefficients.

    • Sum of (3x2+5x−8)+(5x2−13x−5)(3x^2 + 5x - 8) + (5x^2 - 13x - 5):     8x2−8x−138x^2 - 8x - 13

    • Difference of (3−13x−7x2)−(5x2+12x−10)(3 - 13x - 7x^2) - (5x^2 + 12x - 10):     −12x2−25x+13-12x^2 - 25x + 13

    • Simplification of (5x2+13x−4)−(17x2+7x−19)+(5x−7)(3x+1)(5x^2 + 13x - 4) - (17x^2 + 7x - 19) + (5x - 7)(3x + 1):     3x2−10x+83x^2 - 10x + 8

  • Polynomial Multiplication:

    • Multiply each term of the first polynomial by each term of the second polynomial and simplify.

    • Product of (2x−9)(x+6)(2x - 9)(x + 6):     2x2+3x−542x^2 + 3x - 54

    • Product of (2x+5)(7−4x)(2x + 5)(7 - 4x):     −8x2−6x+35-8x^2 - 6x + 35

    • Product of (2x−3)(3x2+2x−1)(2x - 3)(3x^2 + 2x - 1):     6x3−5x2−8x+36x^3 - 5x^2 - 8x + 3

  • Polynomial Classification:

    • Simplifying 4x(x+1)−(3x−8)(x+4)4x(x + 1) - (3x - 8)(x + 4) yields x2+32x^2 + 32, which is classified as a quadratic binomial because it has a degree of 22 and contains 22 terms.

Factoring Polynomials

  • Greatest Common Factor (GCF):

    • An expression has a greatest common factor of 3h3h when 3h3h divides every term evenly, such as 12h+21h2=3h(4+7h)12h + 21h^2 = 3h(4 + 7h).

  • Factoring Quadratic Trinomials:

    • Factoring y2−12y+32y^2 - 12y + 32:     (y−4)(y−8)(y - 4)(y - 8)

    • Factoring perfect square trinomial x2−12x+36x^2 - 12x + 36:     (x−6)2(x - 6)^2

    • Factoring 5x2+17x+65x^2 + 17x + 6:     (5x+2)(x+3)(5x + 2)(x + 3)

    • Factoring 3x2−17x−283x^2 - 17x - 28:     (3x+4)(x−7)(3x + 4)(x - 7)

    • Factoring 3x2−30x−723x^2 - 30x - 72:     3(x−12)(x+2)3(x - 12)(x + 2)

  • Matching Trinomials with Factor Pairs:

    • a2+a−20=(a−4)(a+5)a^2 + a - 20 = (a - 4)(a + 5)

    • a2−9a+20=(a−4)(a−5)a^2 - 9a + 20 = (a - 4)(a - 5)

    • a2−8a−20=(a−10)(a+2)a^2 - 8a - 20 = (a - 10)(a + 2)

    • a2−12a+20=(a−10)(a−2)a^2 - 12a + 20 = (a - 10)(a - 2)

Rational Expressions and Operations

  • Simplifying Rational Expressions:

    • Factor both numerator and denominator completely and cancel common factors.

    • Simplest form of x2+x−2x3−x2+2x−2\frac{x^2 + x - 2}{x^3 - x^2 + 2x - 2}:     x+2x2+2\frac{x + 2}{x^2 + 2}

  • Multiplying and Dividing Rational Expressions:

    • Product of x+24x2+5x+1⋅4x+1x2−4\frac{x + 2}{4x^2 + 5x + 1} \cdot \frac{4x + 1}{x^2 - 4}:     1(x+1)(x−2)\frac{1}{(x + 1)(x - 2)}

    • Quotient of x+3x2−2x−3÷x2+2x−3x+1\frac{x + 3}{x^2 - 2x - 3} \div \frac{x^2 + 2x - 3}{x + 1}:     1x2−4x+3\frac{1}{x^2 - 4x + 3}

    • Quotient of 3x2−27x2x2+13x−7÷3x4x2−1\frac{3x^2 - 27x}{2x^2 + 13x - 7} \div \frac{3x}{4x^2 - 1}:     2x2−17x−9x+7\frac{2x^2 - 17x - 9}{x + 7}

  • Adding and Subtracting Rational Expressions:

    • Sum of 15x−6+7x+6\frac{15}{x - 6} + \frac{7}{x + 6}:     22x+48x2−36\frac{22x + 48}{x^2 - 36}

    • Difference of 5m−m+8m2−4m\frac{5}{m} - \frac{m + 8}{m^2 - 4m}:     4(m−7)m(m−4)\frac{4(m - 7)}{m(m - 4)}

  • Division and Rewriting Form:

    • Rewriting x2+x+4x−2\frac{x^2 + x + 4}{x - 2} via polynomial division:     (x+3)+10x−2(x + 3) + \frac{10}{x - 2}

    • Rewriting x2−x+4x−2\frac{x^2 - x + 4}{x - 2}:     (x+1)+6x−2(x + 1) + \frac{6}{x - 2}

    • Rewriting x2−4x+10x−2\frac{x^2 - 4x + 10}{x - 2}:     (x−2)+6x−2(x - 2) + \frac{6}{x - 2}

    • Rewriting x2−5x+16x−2\frac{x^2 - 5x + 16}{x - 2}:     (x−3)+10x−2(x - 3) + \frac{10}{x - 2}