Identities

Japanese school year: Math II

What you learn

Learn the properties of identities that hold true for all values where both sides are defined, along with methods to determine unknown coefficients. Identities are fundamental for partial fraction decomposition and algebraic manipulation. Understanding polynomial expansion and the difference between equations and identities beforehand is beneficial.

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Key points

This rule states that if $ax + b = 0$ holds for every $x$ , both the coefficient $a$ and the constant $b$ must be $0$ . It is used to determine unknown coefficients in an identity.

ax + b = 0 \iff a = 0, \; b = 0

If the quadratic expression $ax^2 + bx + c = 0$ is true for every $x$ , all coefficients $a, b$ , and $c$ must be $0$ . This serves as a fundamental rule for solving identities.

ax^2 + bx + c = 0 \iff a = 0, \; b = 0, \; c = 0

When two quadratic expressions are equal for all $x$ , matching coefficients ( $a$ and $a'$ , $b$ and $b'$ , $c$ and $c'$ ) are equal. This is used in the method of equating coefficients.

ax^2 + bx + c = a'x^2 + b'x + c' \iff a = a', \; b = b', \; c = c'

After finding coefficients by plugging specific numbers into $x$ , always substitute them back into the original formula to verify that the identity holds for all numbers.

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Practice 1Coming soon