Polynomials

Answer

Step 1: Factor the polynomial. Look for two numbers that multiply to 6 and add to -5. Those are -2 and -3. Step 2: x² - 5x + 6 = (x - 2)(x - 3) Step 3: Set each factor to zero: x - 2 = 0 or x - 3 = 0 …

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Step 1: Find zeros by factoring: 2x² - 7x + 3 = (2x - 1)(x - 3) Zeros: x = 1/2 and x = 3 Step 2: Check sum of zeros: Sum = 1/2 + 3 = 7/2 From formula: -b/a = -(-7)/2 = 7/2 ✓ Step 3: Check product of…

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Given: p(x) = x² - 6x + 8 Compare with ax² + bx + c where a = 1, b = -6, c = 8 Step 1: Sum of zeros formula: α + β = -b/a = -(-6)/1 = 6 Step 2: Product of zeros formula: αβ = c/a = 8/1 = 8 Answer: …

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Step 1: If zeros are α = 3 and β = -2, then: Sum of zeros = α + β = 3 + (-2) = 1 Product of zeros = αβ = 3 × (-2) = -6 Step 2: Use the relationship: α + β = -b/a and αβ = c/a Step 3: Taking a = 1: -…

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Step 1: Factor out common terms: 3x² - 12x = 3x(x - 4) = 0 Step 2: Apply zero product property: Either 3x = 0 or (x - 4) = 0 Step 3: Solve each equation: From 3x = 0: x = 0 From x - 4 = 0: x = 4 St…

Answer

For equal zeros, the discriminant must be zero. For ax² + bx + c, discriminant = b² - 4ac Step 1: Identify coefficients: a = 1, b = k, c = 12 Step 2: Set discriminant = 0: k² - 4(1)(12) = 0 k² - 48 …

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Step 1: Identify the highest power term: The term with highest power is 5x³ Step 2: Find degree: Degree = highest power of x = 3 Step 3: Find leading coefficient: Leading coefficient = coefficient o…

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Given: p(x) = x² - 4x + 3 Step 1: Find p(2): Substitute x = 2: p(2) = (2)² - 4(2) + 3 p(2) = 4 - 8 + 3 = -1 Step 2: Find p(0): Substitute x = 0: p(0) = (0)² - 4(0) + 3 p(0) = 0 - 0 + 3 = 3 Answer: …