Extract common factors, identify identities and distinguish factored expressions from equations.
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Try
A rectangular design has area x² + 5x + 6 square units. Could its side lengths be x + 2 and x + 3? Multiply the proposed lengths before trusting them. Physical lengths impose further restrictions, but the polynomial identity can be checked for every real x.
Predict
The outside product contributes x², the two cross-products contribute 3x and 2x, and the constants contribute 6. Predict which two numbers must be found to factor x² + bx + c into (x + r)(x + s): their sum is b and their product is c.
Experiment
Use a four-cell multiplication table.
Multiply
x
3
x
x²
3x
2
2x
6
Add the cells to recover x² + 5x + 6. Change the expression to x² + x − 6. The factors become (x + 3)(x − 2): the product stays −6 and the middle coefficient becomes 1. Verify all four products.
Understand
Factoring rewrites a sum as a product. First inspect a common factor: 4x² − 12x = 4x(x − 3). Dividing by x while solving would lose a possible zero root; factoring keeps it visible.
Useful identities include a² − b² = (a − b)(a + b) and a² + 2ab + b² = (a + b)². Here a and b can themselves be expressions. A sum of squares does not follow the difference-of-squares rule over the reals.
The zero-product property says that if a product of real or complex numbers equals zero, at least one factor is zero. Thus (x − 4)(x + 2) = 0 gives x = 4 or x = −2. It does not say either factor is zero when the product equals 8. First bring the equation to zero and then factor the resulting expression.
Not every quadratic factors into integer linear factors. Factoring x² − 2 over the reals requires √2. The coefficient domain matters: “cannot factor over the integers” does not mean “has no real roots.” The next lesson supplies a general method.
Apply
For each factorization, multiply back as an independent check. For each equation, ensure the product equals zero before splitting into branches. Distinguish a repeated factor from two distinct solution values.
Continue
Use the quadratic formula when integer factoring is unavailable or uncertain. Completing the square will explain the formula and reveal why some quadratics have no real roots but do have complex roots.
Apply · explain · revise
Practice and reasoning
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Which is a fully factored form over the integers of 6x² − 15x?
Solve x² − x − 12 = 0.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
Solve 2x³ − 18x = 0.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
The identity x² + kx + 12 = (x + 3)(x + 4) holds for all x. Find k.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A learner solves (x − 2)(x + 3) = 6 by setting either factor to zero. Explain the error and solve correctly.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
