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Grade 11 · MCR3U · Functions

MCR3U Equivalent Algebraic Expressions Practice Test (with Solutions)

Equivalent algebraic expressions is the algebra unit of MCR3U: radicals, polynomials and rational expressions rewritten into simpler forms that still equal the original. On Martian Lab it is the Rational Expressions unit, and the free test has 8 questions, from simplifying -2√80 + 2√24 - 3√20 + 3√150 to a rational expression with six restrictions. State the restrictions every time, because tests take marks off when you don't.

Free Rational Expressions Practice Test

8 questions, exactly as a free account gets them in the first Rational Expressions test on Martian Lab. Try each one on paper first, then open the answer.

  1. Factor fully. x2+2xy+y2−z2−2z−1x^{2} + 2xy + y^{2} - z^{2} - 2z - 1
    Show answer
    =x2+2xy+y2−(z2+2z+1)= x^{2} + 2xy + y^{2} - (z^{2} + 2z + 1) =(x+y)2−(z+1)2= \left(x + y\right)^{2} - \left(z + 1\right)^{2} =(x+y−z−1)(x+y+z+1)= \left(x + y - z - 1\right)\left(x + y + z + 1\right)
  2. Simplify the following, show all your work. State restrictions if necessary. −280+224−320+3150-2\sqrt{80} + 2\sqrt{24} - 3\sqrt{20} + 3\sqrt{150}
    Show answer
    =−85+46−65+156= -8\sqrt{5} + 4\sqrt{6} - 6\sqrt{5} + 15\sqrt{6} =196−145= 19\sqrt{6} - 14\sqrt{5}
  3. Simplify the following, show all your work. State restrictions if necessary. (33+42)(3−22)\left(3\sqrt{3} + 4\sqrt{2}\right)\left(\sqrt{3} - 2\sqrt{2}\right)
    Show answer
    =9+46−66−16= 9 + 4\sqrt{6} - 6\sqrt{6} - 16 =−26−7= -2\sqrt{6} - 7
  4. Simplify the following, show all your work. State restrictions if necessary. −3738\dfrac{-3\sqrt{7}}{3\sqrt{8}}
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    =−3738×88= \frac{-3\sqrt{7}}{3\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} =−568= \frac{-\sqrt{56}}{8} =−2148= \frac{-2\sqrt{14}}{8} =−144= \frac{-\sqrt{14}}{4}
  5. Simplify the following, show all your work. State restrictions if necessary. 4x6x2+7x−3−5x2x2+15x+18\dfrac{4x}{6x^2+7x-3} - \dfrac{5x}{2x^2+15x+18}
    Show solution
    =4x(2x+3)(3x−1)−5x(x+6)(2x+3)= \frac{4x}{(2x+3)(3x-1)} - \frac{5x}{(x+6)(2x+3)} =4x(x+6)−5x(3x−1)(2x+3)(3x−1)(x+6)= \frac{4x(x+6) - 5x(3x-1)}{(2x+3)(3x-1)(x+6)} =4x2−24x−15x2+5x(2x+3)(3x−1)(x+6)= \frac{4x^2 - 24x - 15x^2 + 5x}{(2x+3)(3x-1)(x+6)} =−11x2−21x(2x+3)(3x−1)(x+6)= \frac{-11x^2 - 21x}{(2x+3)(3x-1)(x+6)} x≠−32,13,−6x \neq \dfrac{-3}{2}, \dfrac{1}{3}, -6
  6. Simplify the following, show all your work. State restrictions if necessary. −2x2+x+3x2−1×x2+x−22x2+x−6\dfrac{-2x^2+x+3}{x^2-1} \times \dfrac{x^2+x-2}{2x^2+x-6}
    Show answer
    =(−2x+3)(x−1)(x−1)(x+1)×(x+2)(x−1)(2x−3)(x+2)= \frac{(-2x+3)(x-1)}{(x-1)(x+1)} \times \frac{(x+2)(x-1)}{(2x-3)(x+2)} =−1= -1 x≠1,−1,−2,32x \neq 1, -1, -2, \frac{3}{2}
  7. Expand and simplify. 2(3x−4)2−(2x+3)(2x−3)2(3x-4)^2 - (2x+3)(2x-3)
    Show answer
    =2 ⁣(9x2−24x+16)−(4x2−9)= 2\!\left(9x^2 - 24x + 16\right) - \left(4x^2 - 9\right) =18x2−48x+32−4x2+9= 18x^2 - 48x + 32 - 4x^2 + 9 =14x2−48x+41= 14x^2 - 48x + 41
  8. Simplify and state any restrictions on the variable. x−26x2−7x−5÷−3x2+6x3x2+7x−20+2x+34x2−12x+9\frac{x-2}{6x^2-7x-5} \div \frac{-3x^2+6x}{3x^2+7x-20} + \frac{2x+3}{4x^2-12x+9}
    Show solution
    =(x−2)(2x+1)(3x−5)÷−3x(x−2)(x+4)(3x−5)+2x+3(2x−3)2= \frac{(x-2)}{(2x+1)(3x-5)} \div \frac{-3x(x-2)}{(x+4)(3x-5)} + \frac{2x+3}{(2x-3)^2} =(x−2)(2x+1)(3x−5)×(x+4)(3x−5)−3x(x−2)+2x+3(2x−3)2= \frac{(x-2)}{(2x+1)(3x-5)} \times \frac{(x+4)(3x-5)}{-3x(x-2)} + \frac{2x+3}{(2x-3)^2} =(x+4)−3x(2x+1)+2x+3(2x−3)2= \frac{(x+4)}{-3x(2x+1)} + \frac{2x+3}{(2x-3)^2} =(x+4)(2x−3)2−3x(2x+1)(2x+3)−3x(2x+1)(2x−3)2= \frac{(x+4)(2x-3)^2 - 3x(2x+1)(2x+3)}{-3x(2x+1)(2x-3)^2} =(x+4) ⁣(4x2−12x+9)−3x ⁣(4x2+8x+3)−3x(2x+1)(2x−3)2= \frac{(x+4)\!\left(4x^2-12x+9\right) - 3x\!\left(4x^2+8x+3\right)}{-3x(2x+1)(2x-3)^2} =4x3+4x2−39x+36−12x3−24x2−9x−3x(2x+1)(2x−3)2= \frac{4x^3 + 4x^2 - 39x + 36 - 12x^3 - 24x^2 - 9x}{-3x(2x+1)(2x-3)^2} =−8x3−20x2−48x+36−3x(2x+1)(2x−3)2,  x≠0,2,−4,53,−12,32= \frac{-8x^3 - 20x^2 - 48x + 36}{-3x(2x+1)(2x-3)^2},\; x \neq 0, 2, -4, \frac{5}{3}, \frac{-1}{2}, \frac{3}{2}

This Unit on Martian Lab

The lessons cover radicals, inequalities, an introduction to rational expressions, adding and subtracting them, and applications.

  1. Radicals
  2. Inequality
  3. Introduction to Rational Expressions
  4. Adding & Subtracting Rational Expressions
  5. Applications of Rational Expressions

Then 4 full Rational Expressions practice tests to check yourself.

Start Rational Expressions free

Rational Expressions FAQ

What is on an MCR3U equivalent algebraic expressions test?+

Simplifying and multiplying radicals, rationalizing denominators, expanding and factoring polynomials, and multiplying, dividing, adding and subtracting rational expressions with their restrictions. The free test on this page has each of these.

How do you simplify a radical?+

Pull out the largest perfect square factor: √80 = √16 × √5 = 4√5. Then collect like radicals the way you collect like terms, so -8√5 - 6√5 = -14√5.

How do you find the restrictions on a rational expression?+

Factor every denominator and set each factor not equal to 0. When you divide, also restrict the zeros of the second fraction's numerator, because it becomes a denominator when you flip it.

How do you rationalize a denominator?+

Multiply the top and bottom by the radical in the denominator. -3√7 / (3√8) becomes -√56 / 8, which simplifies to -√14 / 4.

How do you factor an expression with six terms?+

Look for groups that are perfect squares. x² + 2xy + y² - z² - 2z - 1 is (x + y)² - (z + 1)², a difference of squares, so it factors to (x + y - z - 1)(x + y + z + 1).

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