Worksheet: Zeros of Polynomial Functions

In this worksheet, we will practice finding the set of zeros of a quadratic, cubic, or higher-degree polynomial function.

Q1:

Find, by factoring, the zeros of the function 𝑓(𝑥)=𝑥+2𝑥35.

  • A7,5
  • B7,5
  • C5,7
  • D6,8
  • E5,7

Q2:

What are the zeros of the function 𝑓(𝑥)=2(𝑥1)7?

  • A1+72 and 172
  • B1+72 and 172
  • C172 and 172
  • D1+72 and 172
  • E1+72 and 172

Q3:

Find, by factoring, the zeros of the function 𝑓(𝑦)=𝑦+8𝑦+7.

  • A7,1
  • B8,1
  • C7,1
  • D7,1
  • E1,8

Q4:

Find the set of zeros of the function 𝑓(𝑥)=𝑥17𝑥+16.

  • A{4,1}
  • B{1,4}
  • C{1}
  • D{4}
  • E{4,1,1,4}

Q5:

If the set of zeros of the function 𝑓(𝑥)=𝑥+𝑏𝑥+343 is {8,8}, find the value of 𝑏.

Q6:

Find, by factoring, the zeros of the function 𝑓(𝑥)=9𝑥+9𝑥40.

  • A5,8
  • B53,83
  • C53,83
  • D5,8
  • E53,83

Q7:

Find the set of zeros of the function 𝑓(𝑥)=𝑥𝑥812𝑥81.

  • A{2,9}
  • B{9,9}
  • C{9,2,9}
  • D{9,2,9}
  • E{2,9}

Q8:

Find the set of zeros of the function 𝑓(𝑥)=13(𝑥4).

  • A{4}
  • B13,4
  • C{4}
  • D13,4

Q9:

Find the set of zeros of the function 𝑓(𝑥)=𝑥1𝑥4.

  • A{2,2}
  • B{1}
  • C{2,2}
  • D{2,1,2}
  • E{1}

Q10:

𝑓(𝑥)=4𝑥+𝑏𝑥5𝑥+42, 𝑓(4)=22, and 𝑓(2)=0. Find the other roots of 𝑓(𝑥) and the value of 𝑏.

  • A𝑏=16, 𝑥=32, 𝑥=72
  • B𝑏=16, 𝑥=32, 𝑥=72
  • C𝑏=16, 𝑥=32, 𝑥=72
  • D𝑏=16, 𝑥=32
  • E𝑏=16, 𝑥=32, 𝑥=72

Q11:

Find all the zeros of 𝑓(𝑥)=𝑥+5𝑥9𝑥45 and state their multiplicities.

  • A𝑥=3 with multiplicity 1, 𝑥=5 with multiplicity 1, 𝑥=3 with multiplicity 1
  • B𝑥=3 with multiplicity 2, 𝑥=3 with multiplicity 2
  • C𝑥=3 with multiplicity 1, 𝑥=5 with multiplicity 1
  • D𝑥=5 with multiplicity 1, 𝑥=3 with multiplicity 2

Q12:

Which of the following functions have the same set of zeros?

  • A𝑘(𝑥)=𝑥+10𝑥 and 𝑓(𝑥)=𝑥20𝑥+100𝑥
  • B𝑘(𝑥)=𝑥10𝑥 and 𝑓(𝑥)=𝑥+20𝑥+100𝑥
  • C𝑘(𝑥)=𝑥10𝑥 and 𝑓(𝑥)=𝑥+20𝑥+100𝑥
  • D𝑘(𝑥)=𝑥+10𝑥 and 𝑓(𝑥)=𝑥+20𝑥+100𝑥
  • E𝑘(𝑥)=𝑥+10𝑥 and 𝑓(𝑥)=𝑥20𝑥+100𝑥

Q13:

Find the set of zeros of the function 𝑓(𝑥)=6𝑥𝑥+64.

  • A
  • B{0}
  • C{8}
  • D{0,8,8}
  • E{8,8}

Q14:

The function 𝑓(𝑥)=𝑎𝑥+54𝑥+81 and the function 𝑔(𝑥)=𝑎𝑥+9 have the same set of zeros. Find 𝑎 and the set of zeros.

  • A𝑎=9, 𝑧(𝑓)={3}
  • B𝑎=9, 𝑧(𝑓)={3}
  • C𝑎=3, 𝑧(𝑓)=13
  • D𝑎=3, 𝑧(𝑓)={3}
  • E𝑎=3, 𝑧(𝑓)={3}

Q15:

Find the set of zeros of the function 𝑓(𝑥)=9𝑥+225𝑥.

  • A{5,5,9}
  • B{0,5}
  • C{5,0,5}
  • D{9,5,5}
  • E{5,5}

Q16:

Find the set of zeros of the function 𝑓(𝑥)=7𝑥112𝑥.

  • A{0,4,4}
  • B{7,4,4}
  • C{7,4,4}
  • D{4,4}
  • E{0,4}

Q17:

Given the function 𝑓(𝑥)=𝑥5𝑥+2𝑥+8, determine the vertical and horizontal intercepts.

  • AVertical intercept: 𝑥=1 or 2; horizontal intercept: 𝑦=8
  • BVertical intercept: 𝑥=1,2, or 4; horizontal intercept: 𝑦=8
  • CVertical intercept: 𝑦=0; horizontal intercept: 𝑥=1 or 2
  • DVertical intercept: 𝑦=8; horizontal intercept: 𝑥=1,2, or 4

Q18:

What is the set of zeros of the function 𝑛(𝑥)=𝑥𝑥+76𝑥+7?

  • A{6}
  • B{7}
  • C{7}
  • D{6}
  • E{7}

Q19:

Find the set of zeros of the function 𝑓(𝑥)=𝑥24.

  • A{24}
  • B{24}
  • C
  • D{0,24}
  • E{0,24}

Q20:

Consider the function 𝑘(𝑥)=5𝑥+2𝑥30𝑥88𝑥+40.

Given that one zero of 𝑘(𝑥) is 13𝑖, find all zeros of 𝑘(𝑥) using synthetic division.

  • A13𝑖,1+3𝑖,1,45
  • B13𝑖,1+3𝑖,2,25
  • C13𝑖,1+3𝑖,4,15
  • D13𝑖,1+3𝑖,2,25
  • E13𝑖,1+3𝑖,4,15

Write the linear factorization of 𝑘(𝑥).

  • A𝑘(𝑥)=(𝑥1+3𝑖)(𝑥13𝑖)(5𝑥+4)(𝑥1)
  • B𝑘(𝑥)=(𝑥1+3𝑖)(𝑥13𝑖)(5𝑥+1)(𝑥4)
  • C𝑘(𝑥)=(𝑥1+3𝑖)(𝑥13𝑖)(5𝑥1)(𝑥+4)
  • D𝑘(𝑥)=(𝑥1+3𝑖)(𝑥13𝑖)(5𝑥2)(𝑥+2)
  • E𝑘(𝑥)=(𝑥1+3𝑖)(𝑥13𝑖)(5𝑥+2)(𝑥2)

Q21:

Consider (𝑥)=16𝑥88𝑥+313𝑥348𝑥+117.

Given that one zero of multiplicity 2 of (𝑥) is 34, find all zeros of (𝑥) using synthetic division.

  • A34,23𝑖,2+3𝑖
  • B34,23𝑖,2+3𝑖
  • C34,26𝑖,2+6𝑖
  • D34,26𝑖,2+6𝑖

Write the linear factorization of (𝑥).

  • A(𝑥)=(4𝑥3)(𝑥2+3𝑖)(𝑥23𝑖)
  • B(𝑥)=(4𝑥3)(𝑥+2+3𝑖)(𝑥+23𝑖)
  • C(𝑥)=(4𝑥3)(𝑥+2+6𝑖)(𝑥+26𝑖)
  • D(𝑥)=(4𝑥3)(𝑥2+6𝑖)(𝑥26𝑖)

Q22:

Consider the function 𝑓(𝑥)=𝑥6𝑥+14𝑥32𝑥40.

Given that one zero of 𝑓(𝑥) is 222, find all zeros of 𝑓(𝑥) using synthetic division.

  • A222, 2+22, 13𝑖, 13𝑖
  • B222, 222, 13𝑖, 13𝑖
  • C222, 222, 13𝑖, 1+3𝑖
  • D222, 2+22, 13𝑖, 1+3𝑖
  • E222, 2+22, 1+3𝑖, 1+3𝑖

Write the linear factorization of 𝑓(𝑥).

  • A𝑓(𝑥)=𝑥2+22𝑥222(𝑥1+3𝑖)(𝑥13𝑖)
  • B𝑓(𝑥)=𝑥2+22𝑥222(𝑥+1+3𝑖)(𝑥1+3𝑖)
  • C𝑓(𝑥)=𝑥2+22𝑥222(𝑥+13𝑖)(𝑥13𝑖)
  • D𝑓(𝑥)=𝑥+2+22𝑥2+22(𝑥1+3𝑖)(𝑥+1+3𝑖)
  • E𝑓(𝑥)=𝑥2+22𝑥+2+22(𝑥1+3𝑖)(𝑥13𝑖)

Q23:

Consider 𝑔(𝑥)=𝑥+6𝑥+38𝑥+24𝑥+136.

Given that one zero of 𝑔(𝑥) is 3+5𝑖, find all zeros of 𝑔(𝑥) using synthetic division.

  • A3+5𝑖, 35𝑖, 2, 2
  • B3+5𝑖, 35𝑖, 2
  • C3+5𝑖, 35𝑖, 2𝑖, 2𝑖
  • D3+5𝑖, 3+5𝑖, 2𝑖, 2𝑖

Write the linear factorization of 𝑔(𝑥).

  • A𝑔(𝑥)=(𝑥+35𝑖)(𝑥+3+5𝑖)(𝑥+2)(𝑥2)
  • B𝑔(𝑥)=(𝑥35𝑖)(𝑥+35𝑖)(𝑥+2𝑖)(𝑥2𝑖)
  • C𝑔(𝑥)=(𝑥+35𝑖)(𝑥+3+5𝑖)(𝑥+2𝑖)(𝑥2𝑖)
  • D𝑔(𝑥)=(𝑥+35𝑖)(𝑥+3+5𝑖)(𝑥2)

Q24:

If 𝐹 is an integral domain and 𝑓(𝑥) is a polynomial with coefficients in 𝐹, then which of the following is true of 𝑓(𝑥)?

  • A𝑓(𝑥) is a constant polynomial.
  • B𝑓(𝑥) is irreducible.
  • C𝑓(𝑥) must have a unique factorization.
  • D𝑓(𝑥) is either irreducible or factorable.

Q25:

Given that 𝑓(𝑥)=𝑥+3𝑥13𝑥15 and 𝑓(1)=0, find the other roots of 𝑓(𝑥).

  • A𝑥=3, 𝑥=5
  • B𝑥=2, 𝑥=6
  • C𝑥=3, 𝑥=5
  • D𝑥=2, 𝑥=6
  • E𝑥=3, 𝑥=5

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