EXERCISES AND PROBLEMS

Finding Patterns And Generalising

Observe, check and compare:

Can you add another row to this number triangle? (It is called Pascal’s triangle).

Read And Learn

Cutting up

Cutting this trapezium into two or three equal pieces is very simple, but cutting it into four is much harder. Try it.

Think And Express Yourself

How interesting!

Think of any three digits that are not all the same

Can you expand the polynomial for (a + b )5 without multiplying the binomial (a + b ) by itself five times?

• Check that the difference is always a multiple of 9 and 11.

• Using algebraic language, show that the observation above is true for any three digits, x, y, z, as long as at least two of them are different.

Tip:

For example, 5, 8 and 3.

SELF-ASSESSMENT

1 Break down the following polynomials into factors using notable products.

a) xx21 –42 + b) yxyx 9124 –22 + c) x 16 –4 d) y 81 –4

2 Calculate the values a, b and c to verify the following equality.

() ( ax xb cx x 32 415 – –) 2 += +

3 Multiply each expression by the LCM of the denominators and simplify.

() () () () () xx xx x 3 21 8 31 12 23 23 ––2 + + +

4 Find the quotient and the remainder of this division: (3x 4 – 5x 3 + 4x 2 – 1) : (x 2 + 2)

5 Calculate the value of m that makes the polynomial P (x) = 7x 3 – mx 2 + 3x – 2 divisible by x + 1.

6 Break these polynomials down into factors: a) x 4 – 12x 3 + 36x 2 b) 2x 3 + 5x 2 – 4x – 3

7 Take out the common factor in each expression: a) (x – 2)(2x + 3) – (5 – x)(x – 2) b) (x + 5)(2x – 1) + (x – 5)(2x – 1)

8 Calculate and simplify where possible. a)

9 Given the polynomial P (x) = (x – 1)2(x – 3): a) Invent a second-degree polynomial Q (x) such that P (x) and Q (x) are relatively prime. b) Find a polynomial R (x) for which this is true:

GCD [P (x), R (x)] = (x – 1)(x – 3)

LCM [P (x), R (x)] = (x – 1)2(x – 3)2(x + 2) anayaeducacion.es Solutions to these problems.

10 Find the value of a and b so that when we divide x 3 + ax 2 + bx – 4 by x + 1 the remainder is –10, and when we divide it by x – 2 the remainder is 2.

11 We inscribe a rhombus inside this rectangle with sides x and y Write the perimeter of the rhombus as a function of the sides of the rectangle.

12 We mix x kg of a paint that costs €5 /kg with y kg of another paint that costs €3 /kg. What would 1 kg of the mixture cost? Express it as a function of x and y Express it as a function of x and y

13 One leg of a right-angled triangle measures 14 cm. Write the perimeter and area of the triangle as a function of the hypotenuse x.

14 Two towns, A and B, are 60 km apart. A car leaves A for B at a velocity of v. At the same time, another car leaves B for A at a velocity of v + 3. Express the time they take to meet as a function of v.

15 Use algebraic language to express the area of the coloured part of this shape using x and y.

16 True or false?

If P (x) = (x 2 – 5) · (x + 7) 2 then: a) The roots of P (x) are 5 y –7. b) P (x) has a double root. c) P (x) has two irrational roots

LEARNING SITUATION

THINK

Look back at the content you have worked on and plan solutions to the problems that you may find. To do this, download the assessment from anayaeducacion.es, think individually and share in groups.

TEST YOUR SKILLS

Now you can do the skills self-evaluation included in anayaeducacion.es.