Polynomials

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The degree is 3. The degree of a polynomial is the highest power of the variable. Here, x³ has the highest power of 3. The polynomial is also called a cubic polynomial because its degree is 3.

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Step 1: Substitute x = 2 into the polynomial Step 2: p(2) = (2)² - 3(2) - 4 Step 3: p(2) = 4 - 6 - 4 Step 4: p(2) = -6 Answer: p(2) = -6. This means when x = 2, the polynomial's value is -6.

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Step 1: Substitute x = 3 into p(x) Step 2: p(3) = (3)² - 5(3) + 6 Step 3: p(3) = 9 - 15 + 6 Step 4: p(3) = 0 Answer: Yes, x = 3 is a zero of the polynomial because p(3) = 0. (A zero means the polynom…

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Step 1: Factor the quadratic: x² - 7x + 10 = (x - 2)(x - 5) Step 2: Set each factor equal to zero: (x - 2) = 0 → x = 2 (x - 5) = 0 → x = 5 Answer: The zeroes are 2 and 5. Verification: When…

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For a quadratic polynomial ax² + bx + c with zeroes α and β: Sum of zeroes: α + β = -b/a Product of zeroes: α × β = c/a Example: For 2x² - 8x + 6 with zeroes 1 and 3: Sum = 1 + 3 = 4 = -(-8)/2 = 8/2…

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Using the formulas for a quadratic ax² + bx + c: Here: a = 3, b = 5, c = -2 Step 1: Sum of zeroes = α + β = -b/a = -5/3 Step 2: Product of zeroes = αβ = c/a = -2/3 Answer: α + β = -5/3 and αβ = -2/3…

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Step 1: Use the relationship: If zeroes are α and β, then the polynomial is p(x) = a(x - α)(x - β), where a ≠ 0 Step 2: With α = 4, β = -3: p(x) = a(x - 4)(x + 3) Step 3: Taking a = 1: p(x)…

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Step 1: Check x = -2: p(-2) = (-2)² - 3(-2) - 10 = 4 + 6 - 10 = 0 ✓ Step 2: Check x = 5: p(5) = (5)² - 3(5) - 10 = 25 - 15 - 10 = 0 ✓ Step 3: Sum = -2 + 5 = 3 = -(-3)/1 = 3 ✓ Step 4: Product =…