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Mathematics EN 9th grade: Introduction to Polynomials

Monomials and Polynomials

- Monomials

Expression with a single term

$ 3x^2,\ -5x,\ 7 $

- Polynomial

Sum of monomials

$ 3x^2 + 2x - 5 $

2. Sum of polynomials

Like terms are added

✔ Example:

$ (2x + 3) + (x + 5) = 3x + 8 $

3. Product of a monomial by a polynomial Distribute the monomial ✔ Example: $ 2x(3x + 4) = 6x^2 + 8x $

4. Operations with polynomials ✔ Example: $ (x + 2)(x + 3) = x^2 + 5x + 6 $

5. Notable Cases

- Square of a binomial

$ (a + b)^2 = a^2 + 2ab + b^2 $

✔ Exemplo:

$ (x + 2)^2 = x^2 + 4x + 4 $

- Diferença de quadrados

$ a^2 - b^2 = (a - b)(a + b) $

✔ Example:

$ x^2 - 9 = (x - 3)(x + 3) $

6. Polynomial Factorization

- Distributive Property

$ ax + ay = a(x + y) $

✔ Example:

$ 3x + 6 = 3(x + 2) $

- Difference of Squares

$ x^2 - 16 = (x - 4)(x + 4) $

- Binomial Square

✔ Example:

$ x^2 + 4x + 4 = (x + 2)^2 $

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Those who liked Introduction to Polynomials, also liked to explore Mathematics EN 9th grade

Explore more about What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a
Examples:
\( |5| = 5 $
$ |-5| = 5 $
Simétrico
The Simétrico of a number is:
$ -a $
Examples:
$ 3 \rightarrow -3 $
$ -7 \rightarrow 7 $
Adição e subtração
Addition with the same sign
Add the values and keep the sign
Examples:
$ 3 + 5 = 8 $
$ -4 + (-6) = -10 $
Soma com sinais contrários
Subtract the values and take the sign of the number with the greatest absolute valueNL#
Examples:
$ 7 + (-3) = 4 $
$ -8 + 5 = -3 $
Soma de números simétricos
The result is always zero
$ a + (-a) = 0 $
Examples:
$ 5 + (-5) = 0 $
Subtração (diferença)
It is transformed into the sum of the opposite
$ a - b = a + (-b) $
Examples:
$ 7 - 3 = 7 + (-3) = 4 $
$ 5 - (-2) = 5 + 2 = 7 $
Divisão de números racionais
Main rule
Multiply by the reciprocal
$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $
Inverso de um número racional
Swap the numerator and denominator
$ \text{Inverso de } \frac{a}{b} = \frac{b}{a}, \quad a \neq 0 $
Examples:
$ \frac{3}{4} \rightarrow \frac{4}{3} $
$ 2 = \frac{2}{1} \rightarrow \frac{1}{2} $
Important:
$ 0 \text{ não tem inverso} $
Examples de divisão
$ \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} $
TIPS
✔ Same signs → add and keep the sign
✔ Different signs → subtract and take the sign of the number with the greatest absolute value
✔ Subtraction → convert to addition
✔ Division → multiply by the reciprocal
✔ Número + simétrico: $ a + (-a) = 0 $
✔ Valor absoluto: $ |a| \geq 0 $
..

What are rational numbers?
They are numbers that can be written in fraction form:
$ \frac{a}{b}, \quad b \neq 0 $
Examples:
$ \frac{3}{4}, -2, 0{,}5, -\frac{7}{3} $
valor absoluto
The valor absoluto represents the distance to zero:
$ |a| = a $ se $ a \geq 0 $
$ |a| = -a $ se $ a < 0 $
Examples:
$ |5| = 5 $
$ |-5| = 5 $
Simétrico
The ..know more

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Explore more about Word that replaces or accompanies a noun, avoiding repetition and indicating person, possession, quantity, or relation.
Ex.: ele, aquele, meu, ninguém.
Subclasses de pronomes
Pronome pessoal
Refers to the persons in speech: 1ª, 2ª ou 3ª pessoa.
- Formas tónicas: used alone or for emphasis (mim, ti, ele, nós).
- Formas átonas: used attached to the verb (me, te, se, o, a, nos, vos).
Pronome pessoal em adjacência verbal
- Ênclise: after the verb (amar-me-ás).
- Próclise: before the verb (me amarás).
- Mesóclise: in the middle of the verb, used with future tense (amar-me-ás).
Contração de pronomes pessoais
Personal pronouns (pronomes pessoais) can combine with direct (direto) and indirect (indireto) objects into a single form.
me + o = mo / nos + o = no=lo / vos + o = vo-lo
te + o = to / lhe + o = lho / lhes + o = lho
Ex.: Dei um telemóvel à minha irmã.
complemento direto + complemento indireto
Dei-o à minha irmã
Dei-lhe um telemóvel
Dei-lho
Pronome pessoal “se”
It can perform different syntactic functions or refer to certain phrasal structures.
- se reflexo: complemento direto (Ela penteia-se)
- se recíproco: complemento indireto (Eles abraçam-se)
- se impessoal: sujeito indeterminado (Diz-se que está a ficar cego)
- se passivo: partícula apassivante (Vendem-se maçãs.)
Pronome demonstrativo
Indicates position or reference: este, essa, aquele, isto, isso, aquilo.
Pronome possessivo
Indicates possession: meu, teu, seu, nossa, vossos.
Pronome indefinido
Indicates vague quantity or identity: alguém, ninguém, tudo, algum, certo.
Pronome relativo
Links clauses and refers back to a previous term: que, quem, cujo, onde...

Word that replaces or accompanies a noun, avoiding repetition and indicating person, possession, quantity, or relation.
Ex.: ele, aquele, meu, ninguém.
Subclasses de pronomes
Pronome pessoal
Refers to the persons in speech: 1ª, 2ª ou 3ª pessoa.
- Formas tónicas: used alone or for emphasis (mim, ti, ele, nós).
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✔ Frase passiva: the subject receives the action.
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👉 Tip:
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CD da ativa → subject (passive)
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