Algebraic Expressions

Simplifying Algebraic Expressions

  • The lesson focuses on simplifying, multiplying, and dividing rational algebraic expressions.

Key Terms

  • Rational Algebraic Expression: A ratio of two polynomials (numerator and denominator are polynomials).
  • Complex Fraction: A fraction whose numerator, denominator, or both contain rational expressions.

Problem of the Day

  • A rectangular computer screen has a length of x2x^2 units and a width of xx units. Find the area of the screen in simplest form.
    • Area = Length x Width
    • Area = (x2−2x−15)∗(x−5)(x^2 - 2x - 15) * (x - 5) ; Factorizing gives: (x−5)(x+3)∗x(x-5)(x+3) * x so the area is (x−5)(x+3)(x-5)(x+3). Reduce to (x+3)(x+3)

Simplifying Rational Algebraic Expressions

  • A rational algebraic expression is in simplest form when the greatest common factor (GCF) of the numerator and denominator is 1.

  • Simplification involves factoring the numerator and denominator and then dividing out common factors.

  • In general, start simplifying a rational expression by completely factoring both its numerator and its denominator, then dividing out any common factors between them.

  • Example:
    2x+6x2−9=2(x+3)(x−3)(x+3)=2x−3\frac{2x + 6}{x^2 - 9} = \frac{2(x + 3)}{(x - 3)(x + 3)} = \frac{2}{x - 3}

  • The greatest common factor is abbreviated as GCF, with the symbol ع.م.أ

  • Division by zero is undefined, so we assume that all values that make the denominators zero are excluded.

Multiplying and Dividing Rational Algebraic Expressions

  • These operations are similar to multiplying and dividing numerical fractions.

  • To multiply, multiply the numerators and multiply the denominators, then simplify the resulting expression.

  • To divide, multiply by the reciprocal of the divisor.

  • Example:

x−5x×x2x2−2x−15\frac{x - 5}{x} \times \frac{x^2}{x^2 - 2x - 15}

  • Simplify the expression by dividing the numerator and denominator by the greatest common factor, which is (x+3)(x + 3).

Example 1

  • Write the following in the simplest form:

2x−102x2−11x+5\frac{2x - 10}{2x^2 - 11x + 5}

  • Factor the numerator and denominator:

2(x−5)(2x−1)(x−5)\frac{2(x - 5)}{(2x - 1)(x - 5)}

  • Divide the numerator and denominator by (x−5)(x - 5).

  • Simplify: 22x−1\frac{2}{2x - 1}

Example 2

  • Write the following in the simplest form:

x3−2x2+9x−186x3−24x2+24x\frac{x^3 - 2x^2 + 9x - 18}{6x^3 - 24x^2 + 24x}

  • Factor by grouping in the numerator and factoring out common factors in the denominator:

(x3−2x2)+(9x−18)6x(x2−4x+4)=x2(x−2)+9(x−2)6x(x−2)(x−2)\frac{(x^3 - 2x^2) + (9x - 18)}{6x(x^2 - 4x + 4)} = \frac{x^2(x - 2) + 9(x - 2)}{6x(x - 2)(x - 2)}

  • Factor further:

(x2+9)(x−2)6x(x−2)(x−2)\frac{(x^2 + 9)(x - 2)}{6x(x - 2)(x - 2)}

  • Divide out the common factor of (x−2)(x - 2).

  • Simplify: x2+96x(x−2)\frac{x^2 + 9}{6x(x - 2)}

Example 3

  • Write the following in the simplest form:

1−u2u2+4u−5\frac{1 - u^2}{u^2 + 4u - 5}

  • Factor the numerator and denominator:

(1−u)(1+u)(u−1)(u+5)\frac{(1 - u)(1 + u)}{(u - 1)(u + 5)}

  • Note that (1−u)=−(u−1)(1 - u) = -(u - 1), so

−(u−1)(1+u)(u−1)(u+5)\frac{-(u - 1)(1 + u)}{(u - 1)(u + 5)}

  • Divide out the common factor of (u−1)(u - 1).

  • Simplify: −(u+1)u+5\frac{-(u + 1)}{u + 5}

Remember

  • Factoring polynomials with four or more terms can be done using the grouping method.
  • Factoring out (-1) from the numerator or denominator can simplify the expression.

Multiplying Rational Algebraic Expressions

  • Multiply numerators and denominators, then simplify.

  • If a, b, c, and d are polynomials where b≠0b \neq 0 and d≠0d \neq 0, then:

ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

  • Example:

3xy+2×2xy=6x2y2+2y\frac{3x}{y + 2} \times \frac{2x}{y} = \frac{6x^2}{y^2 + 2y}

Example 1

  • Write the following in the simplest form:
    12ac15b×5ab26c2\frac{12ac}{15b} \times \frac{5ab^2}{6c^2}

  • Resolving into factors:

2×6×a×c3×5×b×5×a×b×b6×c×c\frac{2 \times 6 \times a \times c}{3 \times 5 \times b} \times \frac{5 \times a \times b \times b}{6 \times c \times c}

  • Dividing by common factors gives: 2a2b3c\frac{2a^2b}{3c}

Important

  • Simplify by dividing out common factors before performing multiplication to make calculations easier.

Example 2

  • Write the following in the simplest form:
    x2+x−6x2+6x+9×x+3x2−6x+8\frac{x^2 + x - 6}{x^2 + 6x + 9} \times \frac{x + 3}{x^2 - 6x + 8}

  • Factor:
    (x+3)(x−2)(x+3)(x+3)×x+3(x−2)(x−4)\frac{(x + 3)(x - 2)}{(x + 3)(x + 3)} \times \frac{x + 3}{(x - 2)(x - 4)}

  • Divide out common factors:

1x−4\frac{1}{x - 4}

Real-Life Example

  • Architectural Engineering: Find the area of a rectangular house plan with length l and width w:

l=2x2+2xx2+5x+6,w=x2−x−64x3l = \frac{2x^2 + 2x}{x^2 + 5x + 6}, w = \frac{x^2 - x - 6}{4x^3}

  • The area is calculated as: A=l×wA = l \times w

  • Substituting expressions for l and w and simplifying leads to:

(x+1)(x−3)2x2(x+3)\frac{(x + 1)(x - 3)}{2x^2(x + 3)}

Example 3

  • Coffee: A company packages coffee in boxes with dimensions given in terms of x. Find the volume of the coffee box in simplest form.

Volume=lwh=1x×x2+x−122x−6×(x+5)Volume = lwh = \frac{1}{x} \times \frac{x^2 + x - 12}{2x - 6} \times (x + 5)

Dividing Rational Algebraic Expressions

  • Similar to dividing numerical fractions: multiply by the reciprocal of the divisor.

  • If a, b, c, and d are polynomials where b≠0,c≠0b \neq 0, c \neq 0, and d≠0d \neq 0, then:

ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

  • Example:
    4xy÷5y+1=4x(y+1)5y\frac{4x}{y} \div \frac{5}{y + 1} = \frac{4x(y + 1)}{5y}

Remember

  • If the product of two numbers is 1, then each is the multiplicative inverse (reciprocal) of the other.

Think

  • Why is it required that a≠0a \neq 0?

Example 1

  • Write the following in the simplest form:
    24x2y5c2d16xy3÷10c2d21\frac{24x^2y^5c^2d}{16xy^3} \div \frac{10c^2d^2}{1}

  • Multiply by the reciprocal:
    24x2y5c2d16xy3×110c2d2\frac{24x^2y^5c^2d}{16xy^3} \times \frac{1}{10c^2d^2}

  • Factor & divide out common factors:

3xdy2\frac{3xd}{y^2}

Complex Fractions

  • A complex fraction is a fraction containing rational expressions in its numerator, denominator, or both.

  • Examples:
    x4y,a−64a,y+1y−8y−75,2d+810d+8\frac{\frac{x}{4}}{y}, \frac{\frac{a - 6}{4}}{a}, \frac{\frac{y + 1}{y - 8}}{\frac{y - 7}{5}}, \frac{\frac{2}{d + 8}}{\frac{10}{d + 8}}

Example 2

  • Write the following in the simplest form:
    x2−36y2+3y−4x2−9x+188y+32\frac{\frac{x^2 - 36}{y^2 + 3y - 4}}{\frac{x^2 - 9x + 18}{8y + 32}}

  • Multiply by reciprocal:
    x2−36y2+3y−4×8y+32x2−9x+18\frac{x^2 - 36}{y^2 + 3y - 4} \times \frac{8y + 32}{x^2 - 9x + 18}

  • Factor & divide out common factors:

8(x+6)(y−1)(x−3)\frac{8 (x + 6)}{(y - 1)(x - 3)}

Steps for Simplifying Complex Fractions

Step 1: Write both the numerator and denominator as single fractions, if needed.

Step 2: Write the complex fraction as a division problem.

Step 3: Multiply the dividend by the reciprocal of the divisor.

Step 4: Divide out common factors and simplify.

Example 1

  • Expression:

a2−b2a2−25b−aa+5\frac{\frac{a^2 - b^2}{a^2 - 25}}{\frac{b - a}{a + 5}}

  • Step 1 & 2: Write the complex fraction as a division problem:
    a2−b2a2−25÷b−aa+5\frac{a^2 - b^2}{a^2 - 25} \div \frac{b - a}{a + 5}

  • Multiply by reciprocal:
    a2−b2a2−25×a+5b−a\frac{a^2 - b^2}{a^2 - 25} \times \frac{a + 5}{b - a}

  • Divide out common factors to simplify: −(a+b)a−5\frac{-(a + b)}{a - 5}

Finding the Least Common Multiple (LCM)

  • The lesson focuses on finding the LCM of algebraic expressions and using it to add and subtract rational expressions.

Problem

  • An aquarium is open at the top and shaped like a rectangular prism with dimensions as shown. Find the surface area of the glass in simplest form.

Finding the Least Common Multiple (LCM) for Algebraic Expressions

  • The LCM of two or more numbers is the smallest number that is a multiple of each of them.

  • To find the LCM of two or more algebraic expressions:

    • Factor each expression completely.
    • Write down all the factors, using the highest power of each factor that appears in any of the expressions. This is the LCM.
      *Reminder: Factoring a algebraic term completely means to express as a product of prime numbers and variable, each raised to its lowest power, 1.

Example LCM Problem

  • Find the LCM of 6ab,8a3,12ab56ab, 8a^3, 12ab^5
    Step 1: Factorize
    6ab=2×3×a×b6ab = 2 \times 3 \times a \times b
    8a3=23×a38a^3 = 2^3 \times a^3
    12ab5=22×3×a×b512ab^5 = 2^2 \times 3 \times a \times b^5
    Step 2: Multiply the factors with the largest exponent
    LCM=23×3×a3×b5=24a3b5LCM = 2^3 \times 3 \times a^3 \times b^5 = 24a^3b^5

Adding and Subtracting Rational Algebraic Expressions

  • Similar to adding and subtracting numerical fractions.

  • To add or subtract rational expressions with like denominators, add or subtract the numerators and write the result over the common denominator. Simplify if possible.

Adding and Subtracting with Unlike Denominators

  • Find the least common multiple (LCM) of the denominators.
    *Multiply:
    *Multiply each fraction in the original expression by an expression of "1" so that each denominator will be the lowest common denominator. You will need to factor common denominators. Rewrite expression.
    *Write the LCM below a single fraction bar. Rewrite the expression inside the fraction bar using the numerators:
    *Combine (add or subtract):
    *When you have combined the numerators over the LCM denominator, then simplify the expression and reduce.

  • Example:

yx(y−1)−1x(y−1)=y−1x(y−1)=1x\frac{y}{x(y - 1)} - \frac{1}{x(y - 1)} = \frac{y - 1}{x(y - 1)} = \frac{1}{x}

Example 1

Simplify: 22x3y3+5b6x2y\frac{2}{2x^3y^3} + \frac{5b}{6x^2y}

Solution: Rewrite terms so there is common denominator 6x2y36x^2y^3

22x3y3<em>3xy3xy+5b6x2y</em>y2y2=6xy+5by26x2y3\frac{2}{2x^3y^3}<em>\frac{3xy}{3xy} + \frac{5b}{6x^2y}</em>\frac{y^2}{y^2} = \frac{6xy+5by^2}{6x^2y^3}

Remember

  • Writing a rational expression in simplest form means dividing the numerator and denominator by their common factors.

Examples

Example 2
Simplify 3x−2x2+4x−12−52x+12\frac{3x - 2}{x^2 + 4x - 12} - \frac{5}{2x + 12}
Factor: 3x−2(x+6)(x−2)−52(x+6)\frac{3x - 2}{(x + 6)(x - 2)} - \frac{5}{2(x + 6)}
Rewrite each term with a common denominator of 2(x+6)(x−2)2(x + 6)(x - 2). Thus, you need to multiply the first term by 2/22/2 and the second term by (x−2)/(x−2)(x-2)/(x-2).
2(3x−2)2(x+6)(x−2)−5(x−2)2(x+6)(x−2)=6x−4−5x+102(x+6)(x−2)=\frac{2(3x - 2)}{2(x + 6)(x - 2)} - \frac{5(x-2)}{2(x + 6)(x - 2)}= \frac{6x-4-5x+10}{2(x + 6)(x - 2)} =
Simplified: 12(x−2)\frac{1}{2(x - 2)}

Real world Example

Maysaa is designing a rectangular sticker for a recycling campaign. The sticker has dimensions: length =y+5y−4\frac{y + 5}{y - 4}, width = yy+5\frac{y}{y + 5}
To find the amont of border trim, find the perimeter P.
To find that, you must add up all of the side lengths (Perimeter) and simplify.
P= 2(length+width)
Write each part with the common denominator: (y−4)(y+5)(y - 4)(y+ 5)

Simplifying Complex Fractions Continued

Step 1
Simplify the numerator and denominator so each has on fraction bar.
*Step 2 *
Multiple the original numerator term by reciprocal of denominator term.
Step 3
Solve and simplify.
Solution:
With the complex fraction: (xy−11x+2\frac{\frac{x}{y}-1}{\frac{1}{x}+2}

  • Simplify the fractions on top & bottom.
    Top: (x/y)-1 = (x−y)y\frac{(x-y)}{y} (Step 1) Bottom: (1/x)+2. = Add one fraction= (1+2x)x\frac{(1+2x)}{x}
  • Rewrite:
    (x−y)/y//(1+2x)/x(x-y)/y // (1+2x)/x
    Then, simplify =
    Combine these two. = (Invert and Multiply)=
    (x−y)/y∗x/(1+2x)(x-y)/y * x/(1+2x)
    SOLUTION ====
    Final Answer:
    (x2−xy)/y+xy(x^2 - xy)/y +xy
    What is LCM?
    The least common denominator, the LCM is used to solve adding and subtracting equations, rational equations and to simplify. LCM can be used to find complex functions so its very important to understand.
  • Review finding LCM with prime factorization of numbers because terms cancel more eaisly.