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Precalculus : Pre-Calculus

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Example Questions

Example Question #1 : Find The Sum And Product Of The Zeros Of A Polynomial

Please choose the best answer from the following choices.

Find the sum and product of the zeros of the following polynomial:

4x^2+15x+11

Possible Answers:

Sum=-15/4

Product=-11/4

Sum=-15/4

Product=11/4

Sum= -7/4

Product=11/4

Sum= 15/4

Product= 11/4

Correct answer:

Sum=-15/4

Product=11/4

Explanation:

Factoring the polynomial will give you (4x+11) and (x+1) .

The zeros of these binomials will be x=-11/4 and x=-1 .

If you add these together (sum) you get -15/4 and if you multiply them together (product) you get 11/4 .

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Example Question #1 : Determine The Number Of Positive And Negative Real Zeros Of A Polynomial Using Descartes' Rule Of Signs

Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs

$y=7x^5-3x^4+2x^3+9x^2-2x+\frac{1}{8}$

Possible Answers:

Positive zeroes: 0,2

Negative zeroes: 1,3

Positive zeroes:

Negative zeroes:

Positive zeroes: 0,2,4

Negative zeroes:

Positive zeroes: 1,3

Negative zeroes: 0,2

Correct answer:

Positive zeroes: 0,2,4

Negative zeroes:

Explanation:

In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.

For the function

$y=7x^5-3x^4+2x^3+9x^2-2x+\frac{1}{8}$

there are four sign changes. As such, the number of positive real zeroes can be

0,2,4.

In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for -x. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.

After substituting we get

$y=-7x^5-3x^4-2x^3+9x^2+2x+\frac{1}{8}$

and there is one sign change.

As such, there can only be one negative root.

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Example Question #1 : Products And Quotients Of Complex Numbers In Polar Form

Find the value of ${}(1+i)^{100}$ ,where the complex number is given by ${}i^2=-1$ .

Possible Answers:

${}i2^{100}$

${}2^{101}$

${}-2^{50}$

${}-i2^{100}$

${}-2^{100}$

Correct answer:

${}-2^{50}$

Explanation:

We note that ${}(1+i)^{2}=1+2i+i^2=1+2i-1=2i$ by FOILing.

We also know that:

$(z)^{nm}=(z^{n})^{m}$

We have by using the above rule: n=2 , m=50

$(1+i)^{100}=((1+i)^{2})^{50}$

Since we know that,

${}(1+i)^{2}=1+2i+i^2=1+2i-1=2i$

We have then:

$(1+i)^{100}=((1+i)^{2})^{50}$ $=(2i)^{50}$

Since we know that:

$(ab)^{n}=a^{n}b^{n}$ , we use a=2 ,b=i

We have then:

$2^{50}({i}^2)^{25}=2^{50}(-1)^{25} =-2^{50}}$

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Example Question #1 : Products And Quotients Of Complex Numbers In Polar Form

Compute the following sum:

${}1+i+i^2+i^3....+i^{2015}$ . Remember {}i is the complex number satisfying ${}i^{2}=-1$ .

Possible Answers:

${}\frac{i}{i-1}$

${}\frac{1}{1-i}$

{}i

{}-i

Correct answer:

Explanation:

Note that this is a geometric series.

Therefore we have:

${}1+i+i^2....+i^{2015}=\frac{1-i^{2016}}{1-i}$

Note that,

${}i^{2016}$ = ${}({i}^{2})^{1008}$ and since ${}i^2=-1$ we have ${}{-1}^{1008}=1$ .

$\frac{1-1}{1-i}=\frac{0}{1-i}=0$

this shows that the sum is 0.

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Example Question #1 : Polar Coordinates And Complex Numbers

Find the following product.

${}(1-i)^{n}(1+i)^n$

Possible Answers:

${}2^{n}$

{}2

${}2^{n-1}$

${}-2i^{n}$

${}2i^{n}$

Correct answer:

${}2^{n}$

Explanation:

Note that by FOILing the two binomials we get the following:

${}(1+i)(1-i)=1-i+i-i^2=2$

Therefore,

${}(1-i)^{n}(1-i)^{n}=2^{n}$

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Example Question #2 : Products And Quotients Of Complex Numbers In Polar Form

Compute the magnitude of ${}(1+i)^{n}$ .

Possible Answers:

${}(1-\sqrt{2})^n$

${}(\sqrt{2})^n+1$

${}(\sqrt{2})^{n+1}$

${}(\sqrt{2})^n$

${}(\sqrt{2})^n-1$

Correct answer:

${}(\sqrt{2})^n$

Explanation:

We have

${}|(1+i)|=\sqrt{2}$ .

We know that ${}|z^n|=|z|^n$

Thus this gives us,

${}|(1+i)^n|=|1+i|^{n}=(\sqrt{2})^{n}$ .

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Example Question #3 : Products And Quotients Of Complex Numbers In Polar Form

Evaluate:

$(2+i)\cdot(1+4i)$

Possible Answers:

i^2

i+4

9i-2

5i-7

Correct answer:

9i-2

Explanation:

To evaluate this problem we need to FOIL the binomials.

(2+i)(1+4i)

=2(1)+2(4i)+1(i)+4i^2

=2+9i+4i^2

Now recall that i^2=-1

Thus,

=2+9i-4

=9i-2

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Example Question #4 : Products And Quotients Of Complex Numbers In Polar Form

Find the product , if

$a=3+7i\:\:\:b=1-i$ .

Possible Answers:

3+4i

10+4i

3-7i

3-3i

Correct answer:

10+4i

Explanation:

To find the product , FOIL the complex numbers. FOIL stands for the multiplication of the Firsts, Outers, Inners, and Lasts.

Using this method we get the following,

ab=(3+7i)(1-i)=3-3i+7i-7i^2=3+4i-7i^2

and because i^2=-1

=3+4i-7(-1)=3+4i+7=10+4i .

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Example Question #6 : Find The Product Of Complex Numbers

Simplify: $i^4 \times i^3$

Possible Answers:

$i+\sqrt{-1}$

Correct answer:

Explanation:

The expression $i^4 \times i^3$ can be rewritten as:

$i^4 \times i^3 = i^2\times i^2\times i^2\times i$

Since $i=\sqrt{-1}$ , the value of i^2 = -1 .

$i^2\times i^2\times i^2\times i = (-1) (-1) (-1)(i) = -i$

The correct answer is:

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Example Question #2 : Polar Coordinates And Complex Numbers

Find the product of the two complex numbers

$\small 3+7i$ and $\small 7+10i$

Possible Answers:

$\small -49+79i$

$\small 21+70i$

$\small 10+17i$

$\small 21+79i+70i^2$

Correct answer:

$\small -49+79i$

Explanation:

The product is

$\small (3+7i)(7+10i)=21+30i+49i+70i^2=21-70+79i=-49+79i$

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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Sum And Product Of The Zeros Of A Polynomial

Example Question #1 : Determine The Number Of Positive And Negative Real Zeros Of A Polynomial Using Descartes' Rule Of Signs

Example Question #1 : Products And Quotients Of Complex Numbers In Polar Form

Example Question #1 : Products And Quotients Of Complex Numbers In Polar Form

Example Question #1 : Polar Coordinates And Complex Numbers

Example Question #2 : Products And Quotients Of Complex Numbers In Polar Form

Example Question #3 : Products And Quotients Of Complex Numbers In Polar Form

Example Question #4 : Products And Quotients Of Complex Numbers In Polar Form

Example Question #6 : Find The Product Of Complex Numbers

Example Question #2 : Polar Coordinates And Complex Numbers

All Precalculus Resources

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