Eleventh Grade (Grade 11) Polynomials and Rational Expressions Questions for Tests and Worksheets

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Grade 11 Polynomials and Rational Expressions

Factor the following polynomial.

$x^3-x^2-8x+12$

$(x+3)(x-2)^2$
$(x+3)(x-2)(x+1)$
$(x-3)(x+1)(x-5)$
$(x-1)(x-2)(x+4)$

Grade 11 Polynomials and Rational Expressions CCSS: HSN-CN.C.9

How many complex roots does the polynomial

$(x-2)^3(x^2+1)$ have? (Remember, real numbers are a subset of complex numbers.)

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.A.1

What is the product

$(x + 5)(x^3 - 2x -3)?$

$x^4 + 5x^3 - 2x^2 - 7x -15$
$x^4 + 5x^3 - 2x^2 - 13x - 15$
$x^4 + 5x^3 - 2x^2 - 10x - 15$
$x^4 + 5x^3 - 2x^2 - 3x - 15$

Grade 11 Polynomials and Rational Expressions CCSS: HSN-CN.C.9

Jeremy is working with the Fundamental Theorem of Algebra, and thinks he's found an exception. Looking at

$f(x) = 4(x-1)^2$ , this will result in only one root,

$x=1$ . Therefore, despite this being a second degree polynomial, there is only one root. Is this correct?

Yes, this is a known exception.
No, this is not a polynomial, it is a quadratic function.
No, if the quadratic formula is used, the other root is found.
No, this root has multiplicity of 2, which means it counts as two roots.

Grade 11 Polynomials and Rational Expressions CCSS: HSN-CN.C.9

Looking at the graph of a quadratic polynomial, roots or zeros correspond to where the graph crosses the x-axis. When the graph just touches the x-axis, this corresponds to a double root. The Fundamental Theorem of Algebra states that a quadratic polynomial will always have 2 roots. How is this reconciled with a quadratic polynomial whose graph does not intersect the x-axis?

Quadratic polynomials always intersect the x-axis.
If a quadratic polynomial doesn't cross the x-axis it is no longer a polynomial, and the Fundamental Theorem of Algebra no longer applies.
When a quadratic polynomial doesn't cross the x-axis, this simply implies that its roots are complex with non-zero imaginary parts.
Simply translate the quadratic polynomial till it does cross the x-axis.

Grade 11 Polynomials and Rational Expressions

Simplify.

$(a - 1)/[(a)^2-1]$

$1/a, \ a!=0$
$1/(a - 1), \ a!= -1$
$1/(1- a), \ a!=1$
$1/(a + 1), \ a!=1$

Grade 11 Polynomials and Rational Expressions

$81 - 9k^2$

$(3k - 9)(3k - 9)$
$(9 + 3k)(9 - 3k)$
$(3k + 9)(3k - 9)$
$(9 + 3k) (9 + 3k)$

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.A.1

Tommy is building a rectangular garden. His dimensions are

$5x$ and

$5x-4$ . Find the area of the rectangular garden he is building in terms of

$x$ . (area = lw)

$25x^2-20x$
$25x^2-4$
$25x^2 +5x - 4$
$25x - 20$

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.D.7

Add and simplify.

$(x+2)/(x-1)+(x-3)/(x+1)$

$(x^2-3x+1)/(x^2-x+4)$
$(2x-1)/(2x)$
$(x^2-x-6)/(x-1)$
$(2x^2-x+5)/((x-1)(x+1))$

Grade 11 Polynomials and Rational Expressions

Factor the following polynomial.

$x^3+3x^2-13x-15$

$(x-5)(x-1)(x+3)$
$(x-5)(x -1)(x-3)$
$(x-2)(x+4)(x+1)$
$(x+5)(x+1)(x-3)$

Grade 11 Polynomials and Rational Expressions

Simplify the rational expression.

$(2y)/(8y^2)$

$y/6$
$(2y)/3$
$1/(4y)$
$2/(5y)$

Grade 11 Polynomials and Rational Expressions

Which expression is not a polynomial?

5
$x^-2+x$
$-3x+5x^14-3$
$x^3-2x^2+3x-2$

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.D.6

Use synthetic division to divide the polynomials:

$(y^2+14y+49)/(y+7)$ .

$y-7$
$y^2+7$
$y+7$
$y+(7)/(y+7)$

Grade 11 Polynomials and Rational Expressions

Use the GCF of the terms to factor the polynomial.

Given:

$23x^4 + 46x^3$

$x^3(23x+46)$
$23x^3(x+2)$
$23x^4(x+2)$
$23x(x^3+2x^2)$

Grade 11 Polynomials and Rational Expressions CCSS: HSN-CN.C.9

What is the maximum number of complex roots with non-zero imaginary part,

$a+bi, b!=0$ , the polynomial

$4x^5 - 5x + 6$ can have?

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.D.6

Use synthetic division to divide the polynomials:

$(x^2+2x-63)/(x+9)$ .

$x-7$
$x+7$
$x^2-7$
$x^2+3x-54$

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.D.7

Add and simplify.

$\ (x^2-4x+3)/(x^2+4x-5)+(x^2+4x+3)/(x^2+6x+5)$

$(2x)/(x+5), \ x!= -1, 1$
$(2x^2+6)/(2x^2+10x), \ x!= 0, -5$
$(10(4x^2+3x))/(24x^2-10x-25)$
$(6)/(x^2+6x+5) , \ x!= 1$

Grade 11 Polynomials and Rational Expressions

To simplify the rational expression

$(x+6)/(x^2+5x-6$ , what is the factored form of the trinomial in the denominator?

$(x+6)(x-1)$
$(x-6)(x+1)$
$(x-6)(x-1)$
$(x+6)(x+1)$

Grade 11 Polynomials and Rational Expressions

Find the value that makes the rational expression undefined.

$8/(y+6)$

$y= -4$
$y=0$
$y=3$
$y=-6$

Grade 11 Polynomials and Rational Expressions CCSS: HSA-APR.D.6

Simplify the rational expression.

$(3x(x-3))/((x-3)(x+3))$

$(x-3)/x, \ \ x!=-3,3$
$(3x)/(x+3), \ \ x!= 3$
$(x+3)/(x-3), \ \ x!=-3$
$1/(x-3), \ \ x!= -3$

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Eleventh Grade (Grade 11) Polynomials and Rational Expressions Questions