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Identities, factor structure and coefficients
Free examples and worked answers. No account needed.
Practise the method.
Identities, factor structure and coefficients
Optional extension. Identities, factor structure and coefficients
Example 1
u+v=7 and uv=12. Find u3+v3 without finding u and v separately.
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Answer: 91
- Expand (u+v)3.
- u3+v3=(u+v)3−3uv(u+v)=73−3×12×7.
Example 2
P(x)=x3+ax2+bx−140. Both x−5 and x−7 are factors. Find the coefficient a.
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Answer: −16
- The third root is 5×7140 because the root product is 140.
- The x2 coefficient is the negative sum of all three roots.
Example 3
u+v=7 and uv=10. Find u3+v3 without finding u and v separately.
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Answer: 133
- Expand (u+v)3.
- u3+v3=(u+v)3−3uv(u+v)=73−3×10×7.
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