Sum and product of roots Flipbook PDF

Sum and product of roots

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Sum and product of roots JHOWIE Y.TIU

Roots of a Polynomial A "root" (or "zero") is where the polynomial is equal to zero:

Put simply: a root is the x-value where the y-value equals zero.

If we have a general polynomial like this: n n-1 n-2 f(x) = ax + bx + cx + z Then: • Adding the roots gives: • Multiplying the roots gives: 𝑏 𝑎

𝑧 𝑎

(for even degree polynomials like quadratics).

𝑧 − (for odd degree polynomials like cubics). 𝑎

Factors •

We can take a polynomial, such as:

f(x) = axn + bxn-1 + cxn-2 + ... + z

•

And then factor it like this:

f(x) = a(x−p)(x−q)(x−r)... •

Then p, q, r, etc are the roots (where the polynomial equals zero)

Quadratic • Let's try this with a Quadratic (where the variable's biggest exponent is 2): ax2 + bx + c

• When the roots are p and q, the same quadratic becomes: a(x−p)(x−q)

• Let's expand a(x−p)(x−q): a(x−p)(x−q)

= a( x2 − px − qx + pq ) = ax2 − a(p+q)x + apq

Now let us compare: Quadratic: ax2 +bx +c

Expanded Factors: ax2 −a(p+q)x+apq

• We can now see that −a(p+q)x = bx, so: −a(p+q) = b p+q = −b/a

• And apq = c, so: pq = c/a

• And we get this result: Adding the roots gives −b/a

Multiplying the roots gives c/a

This can help us answer questions.

• Example: What is an equation whose roots are 5 + √2 and 5 − √

The sum of the roots is (5 + √2) + (5 − √2) = 10

The product of the roots is (5 + √2) (5 − √2) = 25 − 2 = 23

• And we want an equation like: ax2 + bx + c = 0

• When a=1 we can work out that: Sum of the roots = −b/a = -b Product of the roots = c/a = c

• Which gives us this result: x2 − (sum of the roots)x + (product of the roots) = 0

• The sum of the roots is 10, and product of the roots is 23, so we get:

x2 − 10x + 23 = 0

Worked out the examples:

• Find the sum and product of roots of a quadratic equation. x^{2}+4x +5 = 0

• Solution: • Strategy Step-by-Step:Step1: Identify the coefficients a,b,c of the quadratic equation.

• Comparing with the standard form of quadratic equation, we observe that a=1, b=4, and c=5. •

𝑏 Step2: Find the ratio − for the sum of the 𝑎 𝑏 4 Therefore the sum of the roots would be =− . 𝑎 1 𝑐 Step3 :Find the ratio 𝑎 𝑐 5 Product of roots would be = . 𝑎 1

roots.