Understanding Engineering Mathematics

2.1.6 Roots and factors of a polynomial ➤^5276 ➤➤

A.Referring to 2.1.2(iii), what are the (i)factors and (ii)roots of the polynomial
x^3 −x^2 −x+1? What are thesolutionsof the equationx^3 −x^2 −x+ 1 =0?
B.Determine the factors ofx^3 + 2 x^2 − 5 x−6.

2.1.7 Rational functions ➤^5476 ➤➤

Which of the following are rational functions?

(i)

x− 1

x^2 + 1

(ii)

√

x+

1

√

x

(iii)

x− 1

x^2 +x+ 1

(iv)

√

x− 1

x+ 1

2.1.8 Algebra of rational functions ➤^5676 ➤➤

Put the following over a common denominator

(i)

2

x+ 1

−

3

x− 2

(ii)

1

x− 1

+

1

x+ 1

−

1

x+ 2

2.1.9 Division and the remainder theorem ➤^6077 ➤➤

A.Divide 2x^3 +x^2 − 6 x+9byx+2.
B.Use the remainder theorem to find the remainder when 4x^3 − 2 x^2 + 3 x−1 is divided
byx−2.

2.1.10 Partial fractions ➤^6277 ➤➤

Split into partial fractions

x+ 1

(x− 1 )(x+ 3 )

2.1.11 Properties of quadratic expressions and equations

➤ 64 78 ➤➤

A.Solve the quadratic equations

(i)x^2 − 3 x= 0

(ii)x^2 − 5 x+ 6 = 0

(iii) 2x^2 + 3 x− 2 = 0

B. Complete the square for the quadraticx^2 +x+1 and hence determine its minimum

value.

C.What is the (i) sum (ii) product of the roots of the quadraticx^2 + 2 x+3?