This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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QUESTION 1 | Expression | Number of terms | Constant term | Coefficient of x | Degree of the expression | Value of expression if x = -2 and y = -2 | | :-------------------------- | :-------------- | :------------ | :----------------- | :----------------------- | :------------------------------------------- | | 3x^2 + 5 + 6x - 7y | 4 | 5 | 6 | 2 | 19 | | 56xy + 82y^2 - 13x - 15 | 4 | -15 | -13 | 2 | 563 | | 8x^4 + x^2 - 7 | 3 | -7 | 0 | 4 | 125 | Calculations for Question 1: Expression 1: 3x^2 + 5 + 6x - 7y Number of terms: 3x^2, 5, 6x, -7y. There are 4 terms. Constant term: 5. Coefficient of x: 6. Degree of the expression: The highest power is x^2, so the degree is 2. Value if x = -2, y = -2: 3(-2)^2 + 5 + 6(-2) - 7(-2) = 3(4) + 5 - 12 + 14 = 12 + 5 - 12 + 14 = 19 Expression 2: 56xy + 82y^2 - 13x - 15 Number of terms: 56xy, 82y^2, -13x, -15. There are 4 terms. Constant term: -15. Coefficient of x: -13. Degree of the expression: The highest sum of powers in a term is 1+1=2 for 56xy and 2 for 82y^2, so the degree is 2. Value if x = -2, y = -2: 56(-2)(-2) + 82(-2)^2 - 13(-2) - 15 = 56(4) + 82(4) + 26 - 15 = 224 + 328 + 26 - 15 = 552 + 26 - 15 = 578 - 15 = 563 Expression 3: 8x^4 + x^2 - 7 Number of terms: 8x^4, x^2, -7. There are 3 terms. Constant term: -7. Coefficient of x: There is no x term, so the coefficient is 0. Degree of the expression: The highest power is x^4, so the degree is 4. Value if x = -2, y = -2: (Note: y is not in the expression) 8(-2)^4 + (-2)^2 - 7 = 8(16) + 4 - 7 = 128 + 4 - 7 = 132 - 7 = 125 QUESTION 2 2.1 State whether the following expressions are monomial, binomial, or trinomial. 2.1.1 (2x+7) × (3x^2+2x-1) Step 1: Expand the expression. (2x+7)(3x^2+2x-1) = 2x(3x^2+2x-1) + 7(3x^2+2x-1) = 6x^3 + 4x^2 - 2x + 21x^2 + 14x - 7 Step 2: Combine like terms. = 6x^3 + (4x^2+21x^2) + (-2x+14x) - 7 = 6x^3 + 25x^2 + 12x - 7 This expression has 4 terms. The expression is a polynomial (specifically, a quadrinomial). It is not a monomial, binomial, or trinomial. Polynomial 2.1.2 (3p+2) + (2x-1) Step 1: Simplify the expression. (3p+2) + (2x-1) = 3p + 2 + 2x - 1 = 3p + 2x + 1 This expression has 3 terms. The expression is a trinomial. Trinomial 2.1.3 8x^2 + 2x - 5x - 9 Step 1: Simplify the expression by combining like terms. 8x^2 + 2x - 5x - 9 = 8x^2 - 3x - 9 This expression has 3 terms. The expression is a trinomial. Trinomial 2.2 Given that a = -2, b = -3, c = 2 and d = 0, determine the value of the following expression: 2.2.1 (3abc)^d Step 1: Substitute the given values into the expression. (3 × (-2) × (-3) × 2)^0 Step 2: Calculate the value inside the parenthesis. (3 × 6 × 2)^0 (18 × 2)^0 (36)^0 Step 3: Apply the exponent rule x^0 = 1 for any non-zero x. = 1 1 2.2.2 a - 5b^2 + 4ac Step 1: Substitute the given values into the expression. -2 - 5(-3)^2 + 4(-2)(2) Step 2: Perform the exponentiation. -2 - 5(9) + 4(-2)(2) Step 3: Perform the multiplications. -2 - 45 + (-16) -2 - 45 - 16 Step 4: Perform the subtractions. -47 - 16 = -63 -63 2.2.3 36ab - 31bc + 2cd Step 1: Substitute the given values into the expression. 36(-2)(-3) - 31(-3)(2) + 2(2)(0) Step 2: Perform the multiplications in each term. 36(6) - 31(-6) + 0 216 - (-186) + 0 Step 3: Perform the addition. 216 + 186 = 402 402 2.2.4 a^2 + (bc)/(2) - (1)/(3)d Step 1: Substitute the given values into the expression. (-2)^2 + ((-3)(2))/(2) - (1)/(3)(0) Step 2: Perform the exponentiation and multiplications. 4 + (-6)/(2) - 0 Step 3: Perform the division. 4 + (-3) - 0 Step 4: Perform the subtraction. 4 - 3 = 1 1