Question

1. k&u; 3marks take a look at all the terms below. circle all the like terms:

5x² 3y² x² 7
4x y x⁶ -2x²

1. k&u; 4 marks classify the polynomials

(a) 7x³
(b) 4x + 9
(c) 2x² + 5x - 3
(d) 3x⁴ + 2x³ - 5x² + +7x - 9

1. k&u; 13 marks expand and simplify.

a) 3(4x⁴) b) 6x(x² - 2) c) 2(ab² + ab - b²a)
d) -8x²(-4x³ + 1) e) (x - 9)(3x + 4) *bonus 2 marks

Explanation:

3. Identifying Like Terms

• Like terms have the same variable(s) raised to the same power(s).
• For \(5x^{2}\), \(x^{2}\), and \(- 2x^{2}\), the variable is \(x\) and the exponent is \(2\).
• The term \(7\) is a constant. There are no other constant - only terms in the given list.
• \(3y^{2}\) has variable \(y\) (different from \(x\) in \(5x^{2},x^{2},-2x^{2}\)), \(4x\) has \(x\) with exponent \(1\) (different from \(x^{2}\)), \(y\) has variable \(y\) (different from \(x\) terms), \(x^{6}\) has \(x\) with exponent \(6\) (different from \(x^{2}\)).

4. Classifying Polynomials

• Monomial: A polynomial with one term.
• For \(7x^{3}\), it has only one term.
• Binomial: A polynomial with two terms.
• For \(4x + 9\), it has two terms (\(4x\) and \(9\)).
• Trinomial: A polynomial with three terms.
• For \(2x^{2}+5x - 3\), it has three terms (\(2x^{2}\), \(5x\), and \(-3\)).
• Polynomial (or Multinomial): A polynomial with more than three terms.
• For \(3x^{4}+2x^{3}-5x^{2}+7x - 9\), it has five terms.

5. Expanding and Simplifying

a) \(3(4x^{4})\)

• Using the distributive property \(a(b)=ab\)
• \(3\times4x^{4}=12x^{4}\)

b) \(6x(x^{2}-2)\)

• Using the distributive property \(a(b + c)=ab+ac\)
• \(6x\times x^{2}-6x\times2=6x^{3}-12x\)

c) \(2(ab^{2}+ab - b^{2}a)\)

• Using the distributive property \(a(b + c + d)=ab+ac + ad\)
• \(2ab^{2}+2ab-2ab^{2}\)
• Combining like terms: \(2ab^{2}-2ab^{2}+2ab = 2ab\)

d) \(-8x^{2}(-4x^{3}+1)\)

• Using the distributive property \(a(b + c)=ab+ac\)
• \((-8x^{2})\times(-4x^{3})+(-8x^{2})\times1=32x^{5}-8x^{2}\)

e) \((x - 9)(3x + 4)\)

• Using the FOIL method \((a + b)(c + d)=ac+ad+bc+bd\)
• \(x\times3x+x\times4-9\times3x-9\times4\)
• \(3x^{2}+4x-27x - 36\)
• Combining like terms: \(3x^{2}+(4x-27x)-36=3x^{2}-23x - 36\)

Answer:

1. Circle \(5x^{2}\), \(x^{2}\), \(-2x^{2}\)

4.
(a) Monomial
(b) Binomial
(c) Trinomial
(d) Polynomial (or Multinomial)
5.
a) \(12x^{4}\)
b) \(6x^{3}-12x\)
c) \(2ab\)
d) \(32x^{5}-8x^{2}\)
e) \(3x^{2}-23x - 36\)